Coherence diversity in frequency and time

ABSTRACT

Various examples are provided for coherence diversity. In one example, a method includes receiving a product signal transmitted over a plurality of subcarriers, the product signal including a product superposition of a first baseband signal and a second baseband signal; estimating equivalent channel responses for the plurality of subcarriers based upon the pilot symbol in the number of time slots of the plurality of subcarriers; and decoding the second encoded message based at least in part upon the first baseband signal and the equivalent channel responses. The first baseband signal can include a pilot symbol in a number of time slots of at least a portion of the plurality of subcarriers and a first encoded message in a remaining number of time slots of the plurality of subcarriers, and the second baseband signal can include a second encoded message.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims priority to, and the benefit of, U.S.provisional application entitled “Coherence Diversity in Frequency andTime” having Ser. No. 62/443,153, filed Jan. 6, 2017, which is herebyincorporated by reference in its entirety.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

This invention was made with government support under CCF1219065 andCCF1527598 awarded by the National Science Foundation. The governmenthas certain rights in the invention.

BACKGROUND

In practical wireless networks, fading of various links does nottransition in lock step and may not have exactly the same frequencyselectivity. Variations in node mobility and scattering environment mayeasily produce unequal link coherence times. Individual links in awireless network can experience non-identical coherence time andbandwidth due to the differences in mobility or local scattering, apractical scenario where the fundamental limits of communication havebeen mostly unknown. But most investigations either ignore thedifferences in coherence qualities of the links in a network or handleit as a nuisance via worst case analysis.

BRIEF DESCRIPTION OF THE DRAWINGS

Many aspects of the present disclosure can be better understood withreference to the following drawings. The components in the drawings arenot necessarily to scale, emphasis instead being placed upon clearlyillustrating the principles of the present disclosure. Moreover, in thedrawings, like reference numerals designate corresponding partsthroughout the several views.

FIG. 1 provides graphical illustrations of examples of wireless fadingconditions, in accordance with various embodiments of the presentdisclosure.

FIG. 2 provides graphical illustrations of examples of disparity casesbetween two users, in accordance with various embodiments of the presentdisclosure.

FIGS. 3 and 4 are plots illustrating comparisons of sum rate vs.signal-to-noise ratio (SNR), in accordance with various embodiments ofthe present disclosure.

FIG. 5 is a plot illustrating the effect of a user interpolation matrixon the product superposition gain, in accordance with variousembodiments of the present disclosure.

FIG. 6 is a plot illustrating a comparison of system minimum mean squareerror (MMSE) vs. SNR, in accordance with various embodiments of thepresent disclosure.

FIGS. 7-9 are plots illustrating comparisons of sum rate vs.signal-to-noise ratio (SNR), in accordance with various embodiments ofthe present disclosure.

FIG. 10 is a plot illustrating a system rate region for a system withdisparity in both coherence time and coherence bandwidth, in accordancewith various embodiments of the present disclosure.

FIGS. 11 and 12 are tables illustrating degrees of freedom ofblock-fading broadcast channel with no CSI and degrees of freedom ofblock-fading multiple access channel with no CSI, respectively, inaccordance with various embodiments of the present disclosure.

FIG. 13 graphically illustrates three receivers having aligned coherencetimes, in accordance with various embodiments of the present disclosure.

FIGS. 14-16 are plots illustrating examples of degrees of freedom regionof a two-receiver broadcast channel with heterogeneous coherence times,in accordance with various embodiments of the present disclosure.

FIG. 17 is a plot illustrating an example of degrees of freedom regionof a three-receiver broadcast channel with heterogeneous coherencetimes, in accordance with various embodiments of the present disclosure.

FIG. 18 graphically illustrates product superposition transmission forunaligned coherence times, in accordance with various embodiments of thepresent disclosure.

FIG. 19 graphically illustrates blind interference alignment forstaggered coherence times with CSIR, in accordance with variousembodiments of the present disclosure.

FIG. 20 graphically illustrates blind interference alignment with pilottransmission, in accordance with various embodiments of the presentdisclosure.

FIG. 21 graphically illustrates combining blind interference alignmentwith product superposition, in accordance with various embodiments ofthe present disclosure.

FIGS. 22 and 23 are plots illustrating examples of degrees of freedomregion of a two-transmitter multiple access channel with identicalcoherence times, in accordance with various embodiments of the presentdisclosure.

FIGS. 24 and 25 are plots illustrating examples of degrees of freedomregion of a two-transmitter multiple access channel with heterogeneouscoherence times, in accordance with various embodiments of the presentdisclosure.

FIG. 26 is a flow chart illustrating an example of coherence diversitywhich can be implemented by processing circuitry in transceivers,transmitters and/or receivers, in accordance with various embodiments ofthe present disclosure.

FIG. 27 is a schematic block diagram that illustrates an example ofprocessing circuitry employed by transmitters and/or receivers, inaccordance with various embodiments of the present disclosure.

DETAILED DESCRIPTION

Disclosed herein are various embodiments of methods related to coherencydiversity. Wireless nodes can experience non-identical coherence timeand bandwidth due to different node mobility and differences in localscattering. This can lead to a new source of gains in wireless networksentitled coherence diversity, which is made possible by leveraging thedifferences between coherence qualities of the links in the samenetwork. This disclosure analyzes coherence diversity in a wide set ofscenarios. Two wireless nodes experiencing downlink fading that may beeither slow or fast, and either flat or frequency selective, areconsidered. This disclosure explores the nature of coherence diversityand the magnitude of gains whenever the dynamic conditions of twodownlink channels are not identical. The initial analysis is organizedinto three parts: when the disparity is in coherence time, in coherencebandwidth, and in both coherence time and coherence bandwidth. For eachdisparity scenario, a version of product superposition transmissionscheme is highlighted and analyzed, demonstrating the gains inachievable rates. Numerical simulations are presented that shed light onthe behavior of coherence diversity and the advantages of the disclosedschemes. Reference will now be made in detail to the description of theembodiments as illustrated in the drawings, wherein like referencenumbers indicate like parts throughout the several views.

Coherence Diversity for Two-User Broadcast Channel I. Introduction

Variation in coherence time and coherence bandwidth can produceopportunities that are reminiscent of, but quite distinct from, otherforms of diversity like multiuser diversity, and spatial diversity; thisnew phenomenon is called coherence diversity. Unlike multiuser diversity(SNR gains), coherence diversity improves the degrees of freedom(multiplexing gain). Unlike spatial diversity, coherence diversity doesnot necessarily require multiple antennas.

The genesis of coherence diversity goes back to the following: inbroadcasting to a pair of users, one time-selective and one static, itis possible to allow the time-selective user to occupy all the degreesof freedom of its link and still to insert a signal meant for the staticuser in a manner that it produces no interference on the time-selectiveuser. This is made possible by product superposition, a method thatallows the signal of the static user to “disappear” into the equivalentchannel seen by the time-selective user. The main contribution of thisdisclosure is to extend the notion of coherence diversity to generalconditions where either of the two users can be static, time-selective,frequency-selective, or doubly-selective, and show that in all thecombinations where the two channels have non-identical dynamicalcharacteristics, coherence diversity applies and leads to gains, andfurthermore to highlight a methodology to capture those gains andcharacterize the magnitude of such gains.

Results were obtained in the framework of OFDM transmission with pilots.The manifestation of coherence diversity in this context is bestexplained in terms of channel-state information and pilots: in amulti-user OFDM system, pilots resources need to be designed toaccommodate the faster user (in time or frequency), and are thereforewasted for user(s) that need fewer pilots. Product superposition allowssome pilot resources (time/frequency) to be reused to send data to theusers that do not need that pilot resource, without contaminating thosesame pilots for the users that indeed need that pilot.

First consider the case where the disparity of coherence times is largeenough that the overhead of one of the users may be neglected for thepurposes of analysis. This user, which is referred to as “static”(denoted by user s), may have a channel that is either flat orfrequency-selective. A second user (denoted by user d) can be modeledthat is both time-selective and frequency-selective. The system can beanalyzed under a variety of pilot transmission schemes (pilots on allsubcarriers vs. pilot interpolation), as well as channel estimationtechniques (frequency domain versus time domain). In each case a versionof product superposition is disclosed and analyzed, and its gainsdemonstrated.

Second, consider equal coherence time but unequal coherence bandwidth.Third, consider disparity in both coherence time and bandwidth. In eachof these cases a version of product superposition is disclosed andanalyzed. A visual representation of the channel types analyzed in thisdisclosure appears in FIGS. 1 and 2. FIG. 1 illustrates examples ofwireless fading conditions for (a) no selectivity; (b) frequencyselective; (c) time selective; and (d) doubly selective. FIG. 2illustrates examples for disparity cases between two users. Case (a)demonstrates disparity in coherence time with no frequency selectivity.Cases (b) and (c) demonstrate disparity in coherence time. In case (b),user s has no selectivity in time or in frequency, whereas in case (c)user s has frequency selectivity. Cases (d) and (e) demonstratedisparity in coherence bandwidth, where both users have no selectivityin time in case (d), or the same level of time selectivity in case (e).Case (f) demonstrates disparity in both coherence time and coherencebandwidth.

The following notation is used throughout the disclosure: Upper (orlower) letters will be used for frequency-domain (of time domain)symbols; boldface letters will be used for matrices and column vectors.A^(H), A⁻¹, detA, and Tr{A} will denote Hermitian (conjugate transpose),inverse, determinant, and trace, respectively, of matrix A, diag{a}denotes a diagonal matrix whose entries consists of the elements of thevector a, I_(M) denotes the M×M identity matrix, 0_(N×M) denotes the N×Mall-zero matrix, 1_(N) denotes the N all-one vector, and A⊗B denotes theKronecker product of A and B. Furthermore,

^(N×M) is the set of N×M complex matrices,

{⋅} represents expectation, and log is taken to the base 2.

II. System Model

Consider an OFDM downlink transmission to two users where thetransmitter (or a base station) is equipped with M antennas, the firstuser (denoted by user d) is equipped with N_(d) antennas, and the seconduser (denoted by user s) is equipped with N_(s) antennas. The usersfading channels are modeled to be block fading where user d has T_(d)coherence time, and K_(d) coherence bandwidth, whereas user s has T_(s)coherence time, and K_(s) coherence bandwidth. The OFDM downlinktransmission can be described by a two-dimensional lattice (a.k.a.resource block) in time and frequency, where each cell (a.k.a. resourceelement) in the resource block corresponds to one time slot and onesubcarrier. For a resource block having T time slots and K subcarriers,define Y_(k)∈

^(N) ^(d) ^(×T) to be user d received signal at subcarrier k, hence,

$\begin{matrix}{{Y_{k} = {{\sqrt{\frac{\rho}{M}}H_{k}X_{k}} + W_{k}}},} & (1)\end{matrix}$where X_(k)∈

^(M×T) denotes the transmitted signal at subcarrier k with ρ transmittedpower, i.e.

{X_(k)X_(k) ^(H)}=MI_(M), and W_(k)∈

^(N) ^(d) ^(×T) denotes user d additive noise at subcarrier k. It isassumed that W_(k) have zero-mean unit-variance independentcomplex-Gaussian entries, and hence,

{X_(k)X_(k) ^(H)}=MI_(N) _(d) . Furthermore, H_(k)∈

^(N) ^(d) ^(×M) represents the channel frequency response between thetransmitter and user d at subcarrier k, i.e., H_(k)=F_(k,d)h, where

{H _(k) H _(k) ^(H) }=MF _(k,d) F _(k,d) ^(H) =M∥f _(k,d)∥² I _(N) _(d),  (2)and furthermore, h_(n,m) stays constant during T_(d) time slots.

Furthermore, within the resource block, Z_(k)∈

^(N) ^(s) ^(×T) is defined to be user s received signal at subcarrier k,hence,

$\begin{matrix}{{Z_{k} = {{\sqrt{\frac{\rho}{M}}D_{k}X_{k}} + \Xi_{k}}},} & (3)\end{matrix}$where D_(k)∈

^(N) ^(s) ^(×M) represents the channel frequency response between thetransmitter and user s at subcarrier k, i.e., D_(k)=F_(k,s)d, where

${F_{k,s} = {I_{N_{s}} \otimes f_{k,s}^{H}}},{d = \begin{bmatrix}d_{1,1} & \ldots & d_{1,M} \\\vdots & \ddots & \vdots \\d_{N_{s},1} & \ldots & d_{N_{s},M}\end{bmatrix}},$f_(k,s)∈

^(L) ^(s) ^(×1) is the vector comprising the first L_(s) elements of rowk of the DFT matrix, d_(n,m)∈

^(L) ^(s) ^(×1) is the user s channel impulse response vector betweenthe transmit antenna m, and receive antenna n with L_(s) zero-meanunit-variance independent complex-Gaussian taps, and hence,

{D _(k) D _(k) ^(H) }=MF _(k,s) F _(k,s) ^(H) =M∥f _(k,s)∥² I _(N) _(s),  (4)and furthermore, d_(n,m) stays constant during T_(s) time slots.Furthermore, Ξ_(k)∈

^(N) ^(s) ^(×T) is user s additive noise at subcarrier k whose entrieshave zero-mean unit-variance independent complex-Gaussian distribution.

III. Disparity in Coherence Time

In this section, the downlink transmission for two users is examinedwith disparity in coherence time, i.e. T_(d)≠T_(s), but equality incoherence bandwidth, i.e., K_(s)=K_(d). Consider the case whenT_(s)>>T_(d), i.e., the channel of users stays constant over very longinterval of time compared the interval of user d channel. Hence, thecost of sending pilots to estimate the channel of user s can be ignored,and hence, user s channel is assumed to be known at user s whereas userd channel is assumed to be unknown at any node. Throughout this section,the channels of the two users have the same number of taps (i.e.L_(d)=L_(s)=L). The channel of user s stays the same over a longinterval of time. Hence, the selectivity of user s across frequency doesnot change the analysis and the results of this section.

A. Flat Fading Channel (L=1)

The case of flat-fading channel with disparity in coherence time wasexamined where product superposition transmission can be used to providegain for the system rate compared to the conventional transmission. Inthe sequel, the product superposition transmission that was consideredis expressed. The same notation in Section II is followed after removingthe subscript k, since the channel here is flat fading. Consider thetransmission time is the user d coherence time, i.e. T=T_(d). Thetransmitted signal of the product superposition scheme isX=[√{square root over (M)}V,VU],  (5)where V∈

^(M×M) is a user s data matrix whose elements are i.i.d. complexGaussian with zero mean and 1/M variance, and U∈

^(M×(T) ^(d) ^(−M)) is the user d data matrix whose elements are i.i.d.complex Gaussian with zero mean and unit variance. Therefore the signalreceived at user d, during T_(d) time slots, is

$\begin{matrix}{Y = {{H\lbrack {{\sqrt{\rho_{T}}V},{\sqrt{\frac{\rho\;\delta}{M}}{VU}}} \rbrack} + W}} & (6)\end{matrix}$where ρ_(τ), and ρ_(δ) denote the SNR of the pilot signal, and the datatransmission, respectively. Define G=HV to be the user d equivalentchannel, and hence,

$\begin{matrix}{Y = {{H\lbrack {{\sqrt{\rho_{T}}G},{\sqrt{\frac{\rho\;\delta}{M}}{GU}}} \rbrack} + {W.}}} & (7)\end{matrix}$Therefore, user d can estimate its equivalent channel G during the firstM time slots and then decodes U coherently based on the channelestimate. Thus user d can achieve the rate

$\begin{matrix}{{R_{d} \geq {( {1 - \frac{M}{T_{d}}} )\{ {{\log\;\det\; I_{N_{d}}} + {\rho_{\delta}^{\prime}\hat{G}{\hat{G}}^{H}}} \}}},} & (8)\end{matrix}$where Ĝ is the estimated equivalent channel, and

$\begin{matrix}{\rho_{\delta}^{\prime} = {\frac{\rho_{\tau}\rho_{\delta}M^{2}}{1 + {\rho_{\tau}M} + {\rho_{\delta}N_{d}M}}.}} & (9)\end{matrix}$

On the other hand, user s received signal during the first M time slotsisZ ^((p))=√{square root over (ρ_(τ))}DV+Ξ ^((p)),  (10)where Z^((p))∈

^(N) ^(s) ^(×M) is the corresponding additive Gaussian noise. Having avery long coherence time, user s knows its channel D, hence, it candecode its signal V. As a result, user s can achieve the rate,

$\begin{matrix}{R_{s} \geq {\frac{M}{T_{d}}{\{ {{\log\;\det\; I_{N_{s}}} + {\rho_{\tau}{DD}^{H}}} \}.}}} & (11)\end{matrix}$

Furthermore, user s can achieve higher rates as follows. User s canfirst estimate the product DV during the first M time slots, and thendecode U during the following T_(d)−M time slots. Canceling theinterference caused by U, user s can decode V using the power sentthrough the entire transmission time. As a result, user s can achievethe rate,

$\begin{matrix}{{R_{s} \geq {\frac{M}{T_{d}}\{ {{\log\;\det\; I_{N_{s}}} + {\frac{1}{\{ \lambda_{i}^{- 2} \}}{DD}^{H}}} \}}},} & (12)\end{matrix}$where λ_(i) ⁻² is being any of the unordered eigenvalues of ŨŨ^(H),Ũ=[ρ_(τ)I, ρ_(δ)UU^(H)].

Thus, user s can achieve a nonzero rate “for free” in the sense thatuser d achieves approximately the same rate as in the single userscenario. Therefore, product superposition can provide gain over theconventional transmission,

$\begin{matrix}{{R_{d} \geq {( {1 - \frac{M}{T_{d}}} )\{ {{\log\;\det\; I_{N_{d}}} + {\rho_{\delta}^{\prime}\hat{H}{\hat{H}}^{H}}} \}}},} & (13)\end{matrix}$where Ĥ is the estimated channel of user d. The difference between theabove rate, and that in Eq. (8) is the expectation, since the former iswith respect to Ĥ, and the later is with respect to Ĝ.

B. Frequency Selective Channel (L≥1)

Conventional OFDM Downlink Transmission:

For completeness, start by giving the downlink OFDM transmission foruser d providing the estimated channel, the estimation error, and theachievable rate when the channel frequency response is estimated(frequency-domain channel estimation). Consider a resource block withinT_(d) time slots, and K_(d) subcarriers. For estimating the channel atsubcarrier k={1, . . . , K_(d)}, pilot signals are sent during the firstM resource elements at the subcarrier k so that the receiver canestimate the channel corresponding to the M transmission antennas. Thetransmission signal at subcarrier k=1, . . . , K_(d) isX _(k)=[√{square root over (M)}I _(M) ,U _(k)],  (14)Hence, the received signal during T_(d) time slots is

$\begin{matrix}{Y_{k} = {\lbrack {{\sqrt{\rho_{\tau,k}}H_{k}},{\sqrt{\frac{\rho_{d,k}}{M}}H_{k}U_{k}}} \rbrack + {W_{k}.}}} & (15)\end{matrix}$Hence, the minimum mean square error (MMSE) estimate of the user dchannel at the subcarrier k isĤ _(k)=Σ_(HY,k)Σ_(Y,k) ⁻¹(√{square root over (ρ_(τ,k))}H _(k) +W _(k)^((p))),  (16)where W_(k) ^((p))∈

^(N) ^(d) ^(×M) is the additive noise during the first M time slots ofthe subcarrier k, and furthermore,

$\begin{matrix}{{\sum\limits_{{HY},k}{= {{\{ {H_{k}Y_{k}^{{(p)}^{H}}} \}} = {M\sqrt{\rho_{\tau,k}}{f_{k,d}}^{2}I_{N_{d}}}}}},} & (17) \\{\sum\limits_{Y,k}{= {{\{ {Y_{k}^{(p)}Y_{k}^{{(p)}^{H}}} \}} = {( {{M\;\rho_{\tau,k}{f_{k,d}}^{2}} + M} ){I_{N_{d}}.}}}}} & \; \\{{Hence},} & \; \\{{{\hat{H}}_{k} = {{\gamma_{\tau,{kd}}H_{k}} + {\frac{\gamma_{\tau,{kd}}}{\sqrt{\rho_{\tau,k}}}W_{k}^{(p)}}}},} & (18) \\{where} & \; \\{\gamma_{\tau,{kd}} = {\frac{\rho_{\tau,k}{f_{k,d}}^{2}}{{\rho_{\tau,k}{f_{k,d}}^{2}} + 1}.}} & (19)\end{matrix}$The covariance matrix of the estimated channel can be given by,

$\begin{matrix}{{\sum\limits_{\hat{H},k}{= {{\{ {{\hat{H}}_{k}{\hat{H}}_{k}^{H}} \}} = {{M( {{\gamma_{\tau,{kd}}^{2}{f_{k,d}}^{2}} + \frac{\gamma_{\tau,{kd}}^{2}}{\rho_{\tau,k}}} )}I_{N_{d}}}}}},} & (20)\end{matrix}$and furthermore, define

$\begin{matrix}{\sigma_{\hat{H},k}^{2} = {{\frac{1}{N_{d}M}T\; r\{ \sum\limits_{\hat{H},k} \}} = {{{\gamma_{\tau,k}^{2}{f_{k}}^{2}} + \frac{\gamma_{\tau,k}^{2}}{\rho_{\tau,k}}} = {\gamma_{\tau,k}{{f_{k,d}}^{2}.}}}}} & (21)\end{matrix}$Therefore, the user d channel estimation error is

$\begin{matrix}{{{\overset{\sim}{H}}_{k} = {{H_{k} - {\hat{H}}_{k}} = {{( {1 - \gamma_{\tau,{kd}}} )H_{k}} - {\frac{\gamma_{\tau,{kd}}}{\sqrt{\rho_{\tau,k}}}W_{k}^{(p)}}}}},} & (22)\end{matrix}$and furthermore, the estimation error covariance matrix is

$\begin{matrix}\begin{matrix}{\Sigma_{\overset{\sim}{H},k} = {\{ {{\overset{\sim}{H}}_{k}{\overset{\sim}{H}}_{k}^{H}} \}}} \\{= {{M{f_{k,d}}^{2}( {1 - \gamma_{\tau,k}} )^{2}I_{N_{d\;}}} + {M\;\frac{\gamma_{\tau,{kd}}^{2}}{\rho_{\tau,k}}I_{N_{d}}}}} \\{= {\frac{M\;\gamma_{\tau,{kd}}}{\rho_{\tau,k}}{I_{N_{d}}.}}}\end{matrix} & (23)\end{matrix}$Hence, the normalized MMSE of the user d channel estimation can bewritten as

$\begin{matrix}{\sigma_{\overset{\sim}{H},k}^{2} = {{\frac{1}{{MN}_{d}}T\; r\{ \sum\limits_{\overset{\sim}{H},k} \}} = {\frac{\gamma_{\tau,{kd}}}{\rho_{\tau,k}}.}}} & (24)\end{matrix}$

After estimating the channel at subcarrier k={1, . . . , K_(d)}, user dcan decode the transmitted data at the resource elements coherently. Thereceived signal of user d during data transmission in the resourceelements at subcarrier k is

$\begin{matrix}{{Y_{k}^{(d)} = {{\sqrt{\frac{\rho_{d,k}}{M}}H_{k}U_{k}} + W_{k}^{(d)}}},} & (25)\end{matrix}$where W_(k) ^((d))∈

^(N) ^(d) ^(×(T) ^(d) ^(−M)) is the corresponding noise. From Eq. (22),

$\begin{matrix}{Y_{k}^{(d)} = {{\sqrt{\frac{\rho_{d,k}}{M}}{\hat{H}}_{k}U_{k}} + {\sqrt{\frac{\rho_{d,k}}{M}}{\overset{\sim}{H}}_{k}U_{k}} + {W_{k}^{(d)}.}}} & (26)\end{matrix}$Therefore, at subcarrier k, user d can achieve the rate

$\begin{matrix}{{R_{d,k} = {{I( {U_{k}; Y_{k}^{(d)} \middle| {\hat{H}}_{k} } )} \geq {\frac{T_{d} - M}{T_{d}}\{ {{\log\;\det\; I_{N_{d}}} + {\frac{\rho_{d,k}}{1 + {\rho_{d,k}\sigma_{H,k}^{2}}}\frac{{\hat{H}}_{k}{\hat{H}}_{k}^{H}}{M}}} \}}}},} & (27)\end{matrix}$where σ_({tilde over (H)},k) ² is given in (24). By defining thenormalized estimated channel

$\begin{matrix}{{{\overset{\sim}{H}}_{k} = \frac{{\hat{H}}_{k}}{\sigma_{\hat{H},k}}},} & (28)\end{matrix}$where σ_(Ĥ,k) is defined in (21). Thus, the achievable rate of user d inT_(d) time slots and K_(d) subcarriers is

$\begin{matrix}{{R_{d} \geq {\frac{T_{d} - M}{T_{d}K_{d}}{\sum\limits_{k = 1}^{K}{\{ {{\log\;\det\; I_{N_{d}}} + {\rho_{H,k}\frac{{\overset{\sim}{H}}_{k}{\overset{\sim}{H}}_{k}^{H}}{M}}} \}}}}},{where}} & (29) \\{\rho_{H,k} = {\frac{\rho_{d,k}\sigma_{\hat{H},k}^{2}}{1 + {\rho_{d,k}\sigma_{\overset{\sim}{H},k}^{2}}}.}} & (30)\end{matrix}$

According to the power optimization provided below, user d can bemaximized when the values for ρ_(d,k), and ρ_(τ,k) are

$\begin{matrix}{{\rho_{d,k} = {( {1 - \alpha_{k}} )\rho\;\frac{T_{d}}{T_{\delta}}}},{\rho_{\tau,k} = {\alpha_{k}\rho\frac{T_{d}}{T_{\tau}}}},} & (31)\end{matrix}$where T_(δ)=T_(d)−M, T_(τ)=M, and

$\begin{matrix}{{\alpha_{k} = {{- \ell} + \sqrt{\ell( {\ell + 1} )}}},{\ell = {\frac{{T_{\delta}T_{\tau}} + {\rho\; T_{d}T_{\tau}{f_{k,d}}^{2}}}{T_{\delta}\rho\; T_{d}{f_{d,k}}^{2}( {1 - \frac{T_{\tau}}{T_{\delta\;}}} )}.}}} & (32)\end{matrix}$

Power Optimization of ρ_(τ,k), and ρ_(d,k):

The optimal values of ρ_(τ,k) and ρ_(d,k) can be found so that the valueof ρ_(H,k) is maximized, which in turn maximizes the user d achievablerate in Eq. (29). From the law of conservation of energy,ρ_(d,k) T _(δ)+ρ_(τ,k) T _(τ) =ρT,  (33)and hence,ρ_(d,k) T _(δ)=(1−α_(k))ρT,ρ _(τ,k) T _(τ)=α_(k) ρT,  (34)where α_(k)∈[0,1] is the power allocation factor. Therefore,

$\begin{matrix}\begin{matrix}{\rho_{H,k} = \frac{\rho_{d,k}\alpha_{k}{f_{k,d}}^{2}}{1 + {\rho_{d,k}\frac{\gamma_{k}}{\rho_{\tau,k}}}}} \\{= \frac{{\alpha_{k}( {1 - \alpha_{k}} )}\frac{( {\rho\; T} )^{2}}{T_{\tau}T_{\delta}}{f_{k,d}}^{4}}{1 + {\alpha_{k}\frac{\rho\; T}{T_{\tau}}{f_{k,d}}^{2}} + {( {1 - \alpha_{k}} )\;\frac{\rho\; T}{T_{\delta\;}}{f_{k,d}}^{2}}}} \\{{= {\frac{\rho\; T{f_{k,d}}^{2}}{( {T_{\delta} - T_{\tau}} )}\frac{\alpha_{k}( {1 - \alpha_{k}} )}{\alpha_{k} + \ell}}},}\end{matrix} & (35) \\{where} & \; \\{\ell = {\frac{{T_{\delta}T_{\tau}} + {\rho\;{TT}_{T}{f_{k,d}}^{2}}}{T_{\delta\;}\rho\; T{f_{k,d}}^{2}( {1 - \frac{T_{\tau}}{T_{\delta}}} )}.}} & (36)\end{matrix}$By taking the first and the second derivatives of ρ_(H,k) with respectto α_(k), it is possible to obtain

$\begin{matrix}{\mspace{20mu}{{\frac{\partial\rho_{H,k}}{\partial\alpha_{k}} = {\frac{\rho\; T}{T_{\delta} - T_{\tau}}\frac{{( {\alpha_{k} + \ell} )( {1 - {2\alpha}} )} - {\alpha_{k}( {1 - \alpha_{k}} )}}{( {\alpha_{k} + \ell} )^{2}}}},\mspace{20mu}{{and}\mspace{14mu}{hence}},}} & (37) \\\begin{matrix}{\frac{\partial^{2}\rho_{H,k}}{\partial\alpha_{k}^{2}} = \frac{\rho\; T}{( {T_{\delta} - T_{\tau}} )( {\alpha_{k} + \ell} )^{4}}} \\{( {{{- 2}( {\alpha_{k} + \ell} )^{3}} - {2( {\alpha_{k} + \ell} )( {{( {\alpha_{k} + \ell} )( {1 - {2\alpha_{k}}} )} - {\alpha_{k}( {1 - \alpha_{k}} )}} )}} )} \\{= {\frac{{- 2}\rho\; T}{( {T_{\delta} - T_{\tau}} )( {\alpha_{k} + \ell} )^{3}}( {( {\alpha_{k} + \ell} )^{2} - \alpha_{k}^{2} - {2\alpha_{k}\ell} + \ell} )}} \\{{= \frac{{- 2}\rho\;{T( {\ell + \ell^{2}} )}}{( {T_{\delta} - T_{\tau}} )( {\alpha_{k} + \ell} )^{3}}},}\end{matrix} & (38)\end{matrix}$which shows the concavity of ρ_(H,k). Therefore, from the KKTconditions, i.e.

${\frac{\partial\rho_{H,k}}{\partial\alpha_{k}} = 0},$and hence, the optimal α_(k) is given byα_(k) =−l+√{square root over (l(l+1))}.  (39)

Product Superposition OFDM Scheme:

In the following, the product superposition transmission can be givenwhen frequency-domain channel estimation is used, and the productsuperposition gain calculated over the conventional transmission. Thetransmitted signal at subcarrier k={1, . . . , K_(d)} can beX _(k)=[√{square root over (M)}V _(k) ,V _(k) U _(k)],  (40)where V_(k)∈

^(M×M) is the user s signal, with

{V_(k)V_(k) ^(H)}=I_(M), and U_(k)∈

^(M×T) ^(d) ^(−M) is the user d signal, with

{U_(k)U_(k) ^(H)}=MI_(M). Hence, the user d received signal atsubcarrier k during the first M resource elements is

$\begin{matrix}\begin{matrix}{Y_{k}^{(\rho)} = {{\sqrt{\rho_{\tau,k}}H_{k}V_{k}} + W_{k}^{(\rho)}}} \\{{= {{\sqrt{\rho_{\tau,k}}G_{k}} + W_{k}^{(p)}}},}\end{matrix} & (41)\end{matrix}$where G_(k)=H_(k)V_(k) is the user d equivalent channel which can beestimated during the first M resource elements. Following the sameanalysis as presented in Section III-B, the MMSE equivalent channelestimate can be given by

$\begin{matrix}{{{\hat{G}}_{k} = {{\gamma_{\tau,{kd}}G_{k}} + {\frac{\gamma_{\tau,{kd}}}{\sqrt{\rho_{\tau,k}}}W_{k}^{(p)}}}},} & (42)\end{matrix}$where γ_(τ,kd) is given in Eq. (19). Furthermore,

$\begin{matrix}{{\sum\limits_{\hat{G},k}{= {{\{ {{\hat{G}}_{k}{\hat{G}}_{k}^{H}} \}} = {{M( {{\gamma_{\tau,{kd}}^{2}{f_{k,d}}^{2}} + \frac{\gamma_{\tau,{kd}}^{2}}{\rho_{\tau,k}}} )}I_{N_{d}}}}}},{and}} & (43) \\{\sigma_{\hat{G},k}^{2} = {{\frac{1}{N_{d}M}{Tr}\{ \sum\limits_{\hat{G},k} \}} = {\gamma_{\tau,{kd}}{{f_{k,d}}^{2}.}}}} & (44)\end{matrix}$Therefore, the user d equivalent channel estimation error is

$\begin{matrix}{{{\overset{\sim}{G}}_{k} = {{G_{k} - {\hat{G}}_{k}} = {{( {1 - \gamma_{\tau,{kd}}} )G_{k}} - {\frac{\gamma_{\tau,{kd}}}{\sqrt{\rho_{\tau,k}}}W_{k}^{(p)}}}}},} & (45)\end{matrix}$and furthermore, the estimation error covariance matrix is

$\begin{matrix}{\sum\limits_{\overset{\sim}{G},k}{= {{\;\{ {{\overset{\sim}{G}}_{k}{\overset{\sim}{G}}_{k}^{H}} \}} = {\frac{M_{\gamma_{\tau,{kd}}}}{\rho_{\tau,k}}{I_{N_{d}}.}}}}} & (46)\end{matrix}$Hence, the normalized MMSE of user d channel estimation can be writtenas

$\begin{matrix}{\sigma_{\overset{\sim}{G},k}^{2} = {{\frac{1}{{MN}_{d}}{Tr}\{ \sum\limits_{\overset{\sim}{G},k} \}} = {\frac{\gamma_{\tau,{kd}}}{\rho_{\tau,k}}.}}} & (47)\end{matrix}$

After estimating the equivalent channel during the first M resourceelements, the user d signal can be sent during the remaining T_(d)−Mresource elements of subcarrier k. The received signal of user d atsubcarrier k is

$\begin{matrix}\begin{matrix}{{Y_{k}^{(d)} = {{\sqrt{\frac{\rho_{d,k}}{M}}G_{k}U_{k}^{(d)}} + W_{k}^{(d)}}},} \\{= {{\sqrt{\frac{\rho_{d,k}}{M}}{\hat{G}}_{k}U_{k}^{(d)}} + {\sqrt{\frac{\rho_{d,k}}{M}}{\overset{\sim}{G}}_{k}U_{k}^{(d)}} + {W_{k}^{(d)}.}}}\end{matrix} & (48)\end{matrix}$Therefore, at subcarrier k, user d can achieve the rate

$\begin{matrix}\begin{matrix}{R_{d,k} = {I( {U_{k}^{(d)}; Y_{k}^{(d)} \middle| {\hat{G}}_{k} } )}} \\{{\geq {\frac{T_{d} - M}{T_{d}}\{ {{\log\;\det\; I_{N_{d}}} + {\frac{\rho_{d,k}}{1 + {\rho_{d,k}\sigma_{\hat{G},k}^{2}}}\frac{{\hat{G}}_{k}{\hat{G}}_{k}^{H}}{M}}} \}}},}\end{matrix} & (49)\end{matrix}$and hence, for K_(d) subcarriers, and T_(d) time slots, user d canachieve the rate

$\begin{matrix}{{R_{d} \geq {\frac{T_{d} - M}{T_{d}K_{d}}{\sum\limits_{k = 1}^{K}{\{ \;{{\log\;\det\; I_{N_{d}}} + {\rho_{G,k}\frac{{\overset{\sim}{G}}_{k}{\overset{\sim}{G}}_{k}^{H}}{M}}} \}}}}},{{{where}\mspace{14mu}{\overset{\sim}{G}}_{k}} = \frac{{\overset{\sim}{G}}_{k}}{\sigma_{\hat{G},k}}},{and}} & (50) \\{\rho_{G,k} = {\frac{\rho_{d,k}\sigma_{\hat{G},k}^{2}}{1 + {\rho_{d,k}\sigma_{\overset{\sim}{G},k}^{2}}}.}} & (51)\end{matrix}$Also the value of optimal power allocation ρ_(d,k), and ρ_(τ,k) in Eq.(31) can be used to maximize the user d product superposition rate inEq. (50).

Furthermore, during the first M resource elements at subcarrier k={1, .. . , K_(d)}, the user s received signal isZ _(k) ^((p))=√{square root over (ρ_(τ,k))}D _(k) V _(k)+Ξ_(k)^((p)),  (52)where Ξ_(k) ^((p))∈

^(N) ^(s) ^(×M) is the corresponding additive Gaussian noise. Knowingits channel, user s, for K_(d) subcarriers and T_(d) time slots, canachieve the rate

$\begin{matrix}{{R_{s} \geq {\frac{M}{T_{d}K_{d}}{\sum\limits_{k = 1}^{K}{\{ {{\log\;\det\; I_{N_{s}}} + {\frac{1}{\{ \lambda_{i}^{- 2} \}}D_{k}D_{k}^{H}}} \}}}}},} & (53)\end{matrix}$where λ_(i) ⁻² can be any of the unordered eigenvalues of U_(k)U_(k)^(H), U_(k)=[ρ_(τ)I, ρ_(δ)U_(k)U_(k) ^(H)].

Thus user s can achieve “for free” rate in Eq. (53) where user dachieves “approximately” the interference-free rate in Eq. (50)providing higher gain over the conventional transmission achievable ratein Eq. (29).

C. Channel Estimation with Interpolation

In frequency-response channel estimation, other than sending pilots ateach subcarrier, pilots can be sent at some subcarriers and the receivercan use interpolation to estimate the channel at the other subcarriersexploiting the correlation of the channel across subcarriers. In thefollowing, start by giving the achievable rate for conventionaltransmission, and proceed to give the system achievable rate obtained bythe product superposition transmission.

Define the set of subcarriers containing pilots to be {k₁, k₂}, andfurthermore, A_(k)∈

^(N) ^(d) ^(×2N) ^(d) to be the user d channel estimation interpolationmatrix at subcarriers k={1, . . . , K_(d)}. Thus, the estimated channelat subcarrier k is

$\begin{matrix}{{{\hat{H}}_{k} = {A_{k}\begin{bmatrix}{\hat{H}}_{k_{1}} \\{\hat{H}}_{k_{2}}\end{bmatrix}}},} & (54)\end{matrix}$where Ĥ_(k) ₁ , and Ĥ_(k) ₂ are the MMSE channel estimates at thesubcarriers k₁, and k₂, respectively, and are given in Eq. (18). Hence,

$\begin{matrix}{{\hat{H}}_{k} = {{A_{k}\begin{bmatrix}{\gamma_{\tau,k_{1}}H_{k_{1}}} \\{\gamma_{\tau,k_{2}}H_{k_{2}}}\end{bmatrix}} + {{A_{k}\begin{bmatrix}{\frac{\gamma_{\tau,k_{1}}}{\sqrt{\rho_{\tau,k_{1}}}}W_{k_{1}}^{(p)}} \\{\frac{\gamma_{\tau,k_{2}}}{\sqrt{\rho_{\tau,k_{2}}}}W_{k_{2}}^{(p)}}\end{bmatrix}}.}}} & (55)\end{matrix}$Therefore, the estimation covariance matrix can be given by

$\begin{matrix}{{\sum\limits_{\hat{H},k}{= {M\;{A_{k}( {{\begin{bmatrix}{\gamma_{\tau,k_{1}}F_{k_{1}}} \\{\gamma_{\tau,k_{2}}F_{k_{2}}}\end{bmatrix}\begin{bmatrix}{\gamma_{\tau,k_{1}}F_{k_{1}}} \\{\gamma_{\tau,k_{2}}F_{k_{2}}}\end{bmatrix}}^{H} + \begin{bmatrix}{\frac{\gamma_{\tau,k_{1}}^{2}}{\rho_{\tau,k_{1}}}I} & 0 \\0 & {\frac{\gamma_{\tau,k_{2}}^{2}}{\rho_{\tau,k_{2}}}I}\end{bmatrix}} )}A_{k}^{H}}}},\mspace{20mu}{and}} & (56) \\{\mspace{79mu}{\sigma_{\hat{H},k}^{2} = {\frac{1}{N_{d}M}{Tr}{\{ \sum\limits_{\hat{H},k} \}.}}}} & (57)\end{matrix}$Furthermore, the estimation error is

$\begin{matrix}{{\overset{\sim}{H}}_{k} = {{( {F_{k} - {A_{k}\begin{bmatrix}{\gamma_{\tau,k_{1}}F_{k_{1}}} \\{\gamma_{\tau,k_{2}}F_{k_{2}}}\end{bmatrix}}} )h} - {{A_{k}\begin{bmatrix}{\frac{\gamma_{\tau,k_{1}}}{\sqrt{\rho_{\tau,k_{1}}}}W_{k_{1}}^{(p)}} \\{\frac{\gamma_{\tau,k_{2}}}{\sqrt{\rho_{\tau,k_{2}}}}W_{k_{2}}^{(p)}}\end{bmatrix}}.}}} & (58)\end{matrix}$The channel taps are uncorrelated, i.e.

{hh^(H)}=MI_(N) _(d) , and hence, the error covariance matrix is

$\begin{matrix}{\sum\limits_{\overset{\sim}{H},k}{= {{M\;( {F_{k} - {A_{k}\begin{bmatrix}{\gamma_{\tau,k_{1}}F_{k_{1}}} \\{\gamma_{\tau,k_{2}}F_{k_{2}}}\end{bmatrix}}} )( {F_{k} - {A_{k}\begin{bmatrix}{\gamma_{\tau,k_{1}}F_{k_{1}}} \\{\gamma_{\tau,k_{2}}F_{k_{2}}}\end{bmatrix}}} )^{H}} + {{{MA}_{k}\begin{bmatrix}{\frac{\gamma_{\tau,k_{1}}^{2}}{\rho_{\tau,k_{1}}}I} & 0 \\0 & {\frac{\gamma_{\tau,k_{2}}^{2}}{\rho_{\tau,k_{2}}}I}\end{bmatrix}}{A_{k}^{H}.}}}}} & (59)\end{matrix}$Furthermore the normalized MMSE is

$\begin{matrix}{\sigma_{\overset{\sim}{H},k}^{2} = {\frac{1}{N_{d}M}{Tr}{\{ \sum\limits_{\overset{\sim}{H},k} \}.}}} & (60)\end{matrix}$Therefore, for K_(d) subcarriers and T_(d) time slots, user d canachieve the rate

$\begin{matrix}{{R_{d} \geq {{\frac{T_{d} - M}{T_{d}K_{d}}\{ {{\log\;\det\; I_{N_{d}}} + {\rho_{H,k_{1}}\frac{{\overset{\sim}{H}}_{k_{1}}{\overset{\sim}{H}}_{k_{1}}^{H}}{M}}} \}} + {\frac{T_{d} - M}{T_{d}K_{d}}\{ {{\log\;\det\; I_{N_{d}}} + {\rho_{H,k_{2}}\frac{{\overset{\sim}{H}}_{k_{2}}{\overset{\sim}{H}}_{k_{2}}^{H}}{M}}} \}} + {\frac{1}{K_{d}}{\sum\limits_{{k = 1},{k \notin {\{{k_{1},k_{2}}\}}}}^{K}{\{ {{\log\;\det\; I_{N_{d}}} + {\rho_{H,k}\frac{{\overset{\sim}{H}}_{k}{\overset{\sim}{H}}_{k}^{H}}{M}}} \}}}}}},} & (61)\end{matrix}$where ρ_(H,k) is given in Eq. (30) with σ_(Ĥ,k) ², andσ_({tilde over (H)},k) ² given in Eq. (57), and Eq. (60), respectively.

Now, the product superposition scheme can be given for the case ofchannel estimation with interpolation. Define V_(k) ₁ , and V_(k) ₂ tobe an independent users signal at the subcarriers k₁, and k₂,respectively. Hence, the transmitted signal at k₁ and k₂ subcarriers,respectively, areX _(k) ₁ =[√{square root over (M)}V _(k) ₁ ,V _(k) ₁ U _(k) ₁ ],X _(k) ₂ =[√{square root over (M)}V _(k) ₂ ,V _(k) ₂ U _(k) ₂ ].  (62)At subcarrier k=k₁, k₂, during the T_(d) time slots, the transmittedsignal isX _(k) =V _(k) U _(k),  (63)with U_(k)∈

^(M×T) ^(d) , and

$\begin{matrix}{{V_{k} = {B_{k}\begin{bmatrix}V_{k_{1}} \\V_{k_{2}}\end{bmatrix}}},} & (64)\end{matrix}$where B_(k)∈

^(M×2M) is the user s signal interpolation matrix. Therefore, the user destimated equivalent channel is

$\begin{matrix}\begin{matrix}{{\hat{G}}_{k} = {A_{k}\begin{bmatrix}{\hat{G}}_{k_{1}} \\{\hat{G}}_{k_{2}}\end{bmatrix}}} \\{{= {\begin{bmatrix}{\gamma_{\tau,k_{1}}G_{k_{1}}} \\{\gamma_{\tau,k_{2}}G_{k_{2}}}\end{bmatrix} + {A_{k}\begin{bmatrix}{\frac{\gamma_{\tau,k_{1}}}{\sqrt{\rho_{\tau,k_{1}}}}W_{k_{1}}^{(p)}} \\{\frac{\gamma_{\tau,k_{2}}}{\sqrt{\rho_{\tau,k_{2}}}}W_{k_{2}}^{(p)}}\end{bmatrix}}}},}\end{matrix} & (65)\end{matrix}$and hence, the user d estimation covariance matrix is

$\begin{matrix}\begin{matrix}{\sum\limits_{\hat{G},k}{= {\{ {{\hat{G}}_{k}{\hat{G}}_{k}^{H}} \}}}} \\{= {A_{k}( {\begin{bmatrix}{\gamma_{\tau,k_{1}}^{2}\{ {G_{k_{1}}G_{k_{1}}^{H}} \}} & 0 \\0 & {\gamma_{\tau,k_{2}}^{2}\{ {G_{k_{2}}G_{k_{2}}^{H}} \}}\end{bmatrix} +} }} \\{{ \begin{bmatrix}{\frac{\gamma_{\tau,k_{1}}^{2}}{\rho_{\tau,k_{1}}}I} & 0 \\0 & {\frac{\gamma_{\tau,k_{2}}^{2}}{\rho_{\tau,k_{2}}}I}\end{bmatrix} )A_{k}^{H}},} \\{{= {{{MA}_{k}( {\begin{bmatrix}{\gamma_{\tau,k_{1}}^{2}{f_{k_{1}}}^{2}} & 0 \\0 & {\gamma_{\tau,k_{2}}^{2}{f_{k_{2}}}^{2}}\end{bmatrix} + \begin{bmatrix}{\frac{\gamma_{\tau,k_{1}}^{2}}{\rho_{\tau,k_{1}}}I} & 0 \\0 & {\frac{\gamma_{\tau,k_{2}}^{2}}{\rho_{\tau,k_{2}}}I}\end{bmatrix}} )}A_{k}^{H}}},}\end{matrix} & (66)\end{matrix}$and furthermore,

$\begin{matrix}{\sigma_{\hat{G},k}^{2} = {\frac{1}{N_{d}M}{Tr}{\{ \sum\limits_{\hat{G},k} \}.}}} & (67)\end{matrix}$Therefore, the user d estimation error at subcarrier k is

$\begin{matrix}\begin{matrix}{{\overset{\sim}{G}}_{k} = {G_{k} - {\hat{G}}_{k}}} \\{= {{H_{k}{B_{k}\begin{bmatrix}V_{k_{1}} \\V_{k_{2}}\end{bmatrix}}} - {A_{k}\begin{bmatrix}{\gamma_{\tau,k_{1}}H_{k_{1}}V_{k_{1}}} \\{\gamma_{\tau,k_{2}}H_{k_{2}}V_{k_{2}}}\end{bmatrix}} - {A_{k}\begin{bmatrix}{\frac{\gamma_{\tau,k_{1}}}{\sqrt{\rho_{\tau,k_{1}}}}W_{k_{1}}^{(p)}} \\{\frac{\gamma_{\tau,k_{2}}}{\sqrt{\rho_{\tau,k_{2}}}}W_{k_{2}}^{(p)}}\end{bmatrix}}}} \\{{= {{( {{H_{k}B_{k}} - {A_{k}\begin{bmatrix}{\gamma_{\tau,k_{1}}H_{k_{1}}} & 0 \\0 & {\gamma_{\tau,k_{2}}H_{k_{2}}}\end{bmatrix}}} )\begin{bmatrix}V_{k_{1}} \\V_{k_{2}}\end{bmatrix}} - {A_{k}\begin{bmatrix}{\frac{\gamma_{\tau,k_{1}}}{\sqrt{\rho_{\tau,k_{1}}}}W_{k_{1}}^{(p)}} \\{\frac{\gamma_{\tau,k_{2}}}{\sqrt{\rho_{\tau,k_{2}}}}W_{k_{2}}^{(p)}}\end{bmatrix}}}},}\end{matrix} & (68)\end{matrix}$and hence, the error covariance matrix is

$\begin{matrix}{{\sum\limits_{\hat{G},k}{= {{\{ {( {{H_{k}B_{k}} - {A_{k}\begin{bmatrix}{\gamma_{\tau,k_{1}}H_{k_{1}}} & 0 \\0 & {\gamma_{\tau,k_{2}}H_{k_{2}}}\end{bmatrix}}} )( {{H_{k}B_{k}} - {A_{k}\begin{bmatrix}{\gamma_{\tau,k_{1}}H_{k_{1}}} & 0 \\0 & {\gamma_{\tau,k_{2}}H_{k_{2}}}\end{bmatrix}}} )^{H}} \}} + {{{MA}_{k}\begin{bmatrix}{\frac{\gamma_{\tau,k_{1}}^{2}}{\rho_{\tau,k_{1}}}I} & 0 \\0 & {\frac{\gamma_{\tau,k_{2}}^{2}}{\rho_{\tau,k_{2}}}I}\end{bmatrix}}A_{k}^{H}}}}},} & (69)\end{matrix}$and furthermore,

$\begin{matrix}{\sigma_{\overset{\sim}{G},k}^{2} = {\frac{1}{N_{d}M}{Tr}{\{ \sum\limits_{\overset{\sim}{G},k} \}.}}} & (70)\end{matrix}$Therefore, the achievable rates of the dynamic and the static users are

$\begin{matrix}{{R_{d} \geq {{\frac{T_{d} - M}{T_{d}K_{d}}\{ {{\log\;\det\; I_{N_{d}}} + {\rho_{G,k_{1}}\frac{{\overset{\_}{G}}_{k_{1}}{\overset{\_}{G}}_{k_{1}}^{H}}{M}}} \}} + {\frac{T_{d} - M}{T_{d}K_{d}}\{ {{\log\;\det\; I_{N_{d}}} + {\rho_{G,k_{2}}\frac{{\overset{\_}{G}}_{k_{2}}{\overset{\_}{G}}_{k_{2}}^{H}}{M}}} \}} + {\frac{1}{K_{d}}{\sum\limits_{{k = 1},{k \notin {\{{k_{1},k_{2}}\}}}}^{K}{\{ {{\log\;\det\; I_{N_{d}}} + {\rho_{G,k}\frac{{\overset{\_}{G}}_{k}{\overset{\_}{G}}_{k}^{H}}{M}}} \}}}}}},} & (71)\end{matrix}$where ρ_(G,k) is given in Eq. (51) with σ_(Ĝ,k) ², andσ_({tilde over (G)},k) ² given in Eq. (67), and Eq. (70),respectively. user s decodes its signal sent on the subcarriers k₁ andk₂ obtaining the rate

$\begin{matrix}{R_{s} \geq {\frac{M}{T_{d}K_{d}}{\sum\limits_{k = {\{{k_{1},k_{2}}\}}}{( {{\log\;{\det I}_{N_{s}}} + {\frac{1}{\{ \lambda_{i}^{- 2} \}}D_{k}D_{k}^{H}}} \}.}}}} & (72)\end{matrix}$

As shown in Eq. (68), the user s signal interpolation matrix affects theerror of the user d channel, σ_({tilde over (G)},k) ². Although linearB_(k) can be used, the system achievable rate can be maximized bydesigning an optimal B_(k) matrix. The optimal B_(k) is

$\begin{matrix}{{B_{k} = {\arg\;\min\;\sigma_{\overset{\sim}{G},k}^{2}}}{{{subject}\mspace{14mu}{to}\mspace{14mu}{Tr}\{ {B_{k}B_{k}^{H}} \}} \leq M}} & (73)\end{matrix}$Using the identity TI{XY}=Tr{YX},

$\begin{matrix}{{\sigma_{\overset{\sim}{G},k}^{2} = {\frac{1}{N_{d}M}{Tr}\{ {{B_{k}^{H}\Psi_{k}B_{k}} - {B_{k}\Phi_{k}} - {\Phi_{k}^{H}B_{k}^{H}} + {{{MA}_{k}( {\begin{bmatrix}{\gamma_{\tau,k_{1}}^{2}{f_{k_{1}}}^{2}} & 0 \\0 & {\gamma_{\tau,k_{2}}^{2}{f_{k_{2}}}^{2}}\end{bmatrix} + \begin{bmatrix}{\frac{\gamma_{\tau,k_{1}}^{2}}{\rho_{\tau,k_{1}}}I} & 0 \\0 & {\frac{\gamma_{\tau,k_{2}}^{2}}{\rho_{\tau,k_{2}}}I}\end{bmatrix}} )}A_{k}^{H}}} \}}},} & (74) \\{\mspace{79mu}{where}} & \; \\{\mspace{79mu}{\Phi_{k} = {\{ {\begin{bmatrix}{\gamma_{\tau,k_{1}}H_{k_{1}}^{H}} & 0 \\0 & {\gamma_{\tau,k_{2}}H_{k_{2}}^{H}}\end{bmatrix}A_{k}^{H}H_{k}} \}}}} & (75) \\{\mspace{79mu}{\Psi_{k} = {\{ {H_{k}^{H}H_{k}} \}.}}} & \;\end{matrix}$Hence,

$\begin{matrix}{B_{k} = {{{\arg\;\min\;{Tr}\{ {B_{k}^{H}\Psi_{k}\; B_{k}} \}} - {{Tr}\{ {B_{k}\Phi_{k}} \}} - {{Tr}\{ {\Phi_{k}^{H}B_{k}^{H}} \}{subject}\mspace{14mu}{to}\mspace{14mu}{Tr}\{ {B_{k}B_{k}^{H}} \}}} \leq {M.}}} & (76)\end{matrix}$The above optimization problem is convex, and furthermore, can be solvedusing CVX, which is a package for specifying and solving convexprograms.

D. Time-Domain Channel Estimation

In the previous discussion, product superposition is applied on the OFDMsystem when the channel estimation is done in the frequency domain,where the channel frequency response, {H_(k)}_(k), is estimated, eitherwith or without interpolation. Now consider applying the productsuperposition when channel estimation is done in the time-domain, wherethe channel impulse response, h, is estimated. The rate achieved in theconventional system when time-domain channel estimation is used is firstconsidered, then the system achievable rate for the productsuperposition scheme is examined. Define, {tilde over (Y)}₂(t)∈

^(K×1) to be the received signal at receive antenna n=1, . . . , N_(d)at time slot t. Hence,

$\begin{matrix}{{{{\overset{\sim}{Y}}_{n}(t)} = {{\sqrt{\frac{\rho}{M}}{\sum\limits_{m = 1}^{M}{{diag}\{ {F\lbrack {h_{n,m}^{H},0_{1 \times {({K - L_{d}})}}^{H}} \rbrack}^{H} \}{{\overset{\sim}{X}}_{m}(t)}}}} + {{\overset{\sim}{W}}_{n}(t)}}},} & (77)\end{matrix}$where F∈

^(K×K) is the DFT matrix, {tilde over (X)}_(m)(t)∈

^(K×1) is the transmitted signal at antenna m=1, . . . , M at time slott, and {tilde over (W)}_(n)(t)∈

^(K×1) is the additive Gaussian noise at receive antenna n at time slott. Therefore,

$\begin{matrix}{{{\overset{\sim}{Y}}_{n}(t)} = {{\sqrt{\frac{\rho}{M}}{\sum\limits_{m = 1}^{M}{{diag}\{ {{\overset{\sim}{X}}_{m}(t)} \}{F\lbrack {h_{n,m}^{H},0_{1 \times {({K - L_{d}})}}^{H}} \rbrack}^{H}}}} + {{{\overset{\sim}{W}}_{n}(t)}.}}} & (78)\end{matrix}$For estimating h_(n,m), at least L pilots are needed for each transmitantenna m=1, . . . , M. Assume that LM pilots are sent at the first Lsubcarriers where M pilots are sent during the first M time slots ofeach subcarrier. Furthermore, assume that

$\begin{matrix}{{\overset{\sim}{X}}_{r}^{(p)} = \{ \begin{matrix}{{\sqrt{M}1_{L}},} & {{r = m},} \\{0_{L \times 1},} & {{r \neq m},}\end{matrix} } & (79)\end{matrix}$is the transmitted pilot signal to estimate h_(n,m), and hence, thereceived signal at antenna n during pilot transmission can be given by{tilde over (Y)} _(n) ^((P))=√{square root over (ρ_(τ))}F _(L) h _(n,m)+{tilde over (W)} _(n) ^((P)),  (80)where F_(L)∈

^(L×L), denotes the first L columns and the first L rows of F, and W_(n)^((P))∈

^(L×1) is the corresponding additive noise.

Using MMSE estimation, the estimated channel isĥ _(n,m)=√{square root over (ρ_(τ))}F _(L) ^(H)(ρ_(τ) F _(L) F _(L) ^(H)+I _(L))⁻¹ {tilde over (Y)} _(n) ^((P)),  (81)and hence,

$\begin{matrix}\begin{matrix}{\sum\limits_{\hat{h}}{= {\{ {{\hat{h}}_{n,m}{\hat{h}}_{n,m}^{H}} \}}}} \\{= {{\rho_{\tau}^{2}{F_{L}^{H}( {{\rho_{\tau}F_{L}F_{L}^{H}} + I_{L}} )}^{- 1}F_{L}{F_{L}^{H}( {{\rho_{\tau}F_{L}F_{L}^{H}} + I_{L}} )}^{- 1}F_{L}} +}} \\{\rho_{\tau}{F_{L}^{H}( {{\rho_{\tau}F_{L}F_{L}^{H}} + I_{L}} )}^{- 2}{F_{L}.}}\end{matrix} & (82)\end{matrix}$The estimation error is {tilde over (h)}_(n,m)=h_(n,m)−ĥ_(n,m), andhence,

$\begin{matrix}\begin{matrix}{\sum\limits_{\overset{\sim}{h}}{= {\{ {{\overset{\sim}{h}}_{n,m}{\overset{\sim}{h}}_{n,m}^{H}} \}}}} \\{= {{\sum\limits_{\hat{h}}{+ I_{L}}} - {2\;\rho_{\tau}{F_{L}^{H}( {{\rho_{\tau}F_{L}F_{L}^{H}} + I_{L}} )}^{- 1}{F_{L}.}}}}\end{matrix} & (83)\end{matrix}$Therefore the estimated channel, and the estimation error at subcarrierk are

$\begin{matrix}{{{\hat{H}}_{k} = {F_{k}\hat{h}}},{\hat{h} = \begin{bmatrix}{\hat{h}}_{1,1} & \ldots & {\hat{h}}_{1,M} \\\vdots & \ddots & \vdots \\{\hat{h}}_{N_{d},1} & \ldots & {\hat{h}}_{N_{d},M}\end{bmatrix}},} & (84) \\{{{\overset{\sim}{H}}_{k} = {F_{k}\overset{\sim}{h}}},{\overset{\sim}{h} = {\begin{bmatrix}{\overset{\sim}{h}}_{1,1} & \ldots & {\overset{\sim}{h}}_{1,M} \\\vdots & \ddots & \vdots \\{\overset{\sim}{h}}_{N_{d},1} & \ldots & {\overset{\sim}{h}}_{N_{d},M}\end{bmatrix}.{Furthermore}}},} & \; \\\begin{matrix}{\sigma_{\hat{H},k}^{2} = {\frac{1}{N_{d}M}{Tr}\{ {\{ {F_{k}\hat{h}{\hat{h}}^{H}F_{k}^{H}} \}} \}}} \\{{= {\frac{1}{N_{d}}{Tr}\{ {{F_{k}( {I_{N_{d}} \otimes \sum\limits_{\hat{h}}} )}F_{k}^{H}} \}}},} \\{\sigma_{\hat{H},k}^{2} = {\frac{1}{N_{d}M}{Tr}\{ {\{ {F_{k}\overset{\sim}{h}{\overset{\sim}{h}}^{H}F_{k}^{H}} \}} \}}} \\{= {\frac{1}{N_{d}}{Tr}{\{ {{F_{k}( {I_{N_{d}} \otimes \sum\limits_{\overset{\sim}{h}}} )}F_{k}^{H}} \}.}}}\end{matrix} & (85)\end{matrix}$Thus user d can achieve the rate

$\begin{matrix}{{R_{d} \geq {{\frac{T_{d} - M}{T_{d}K_{d}}{\sum\limits_{k = 1}^{L}\;{\{ {{\log\;\det\; I_{N_{d}}} + {\rho_{H,k}\frac{{\overset{\_}{H}}_{k}{\overset{\_}{H}}_{k}^{H}}{M}}} \}}}} + {\frac{1}{K_{d}}{\sum\limits_{k = {L + 1}}^{K}\;{\{ {{\log\;\det\; I_{N_{d}}} + {\rho_{H,k}\frac{{\overset{\_}{H}}_{k}{\overset{\_}{H}}_{k}^{H}}{M}}} \}}}}}},} & (86)\end{matrix}$where ρ_(H,k), H _(k) are given in Eq. (30), and Eq. (28), respectively.

Now consider the product superposition scheme in the case of time-domainchannel estimation. Assume the pilot signal in this case is given by

$\begin{matrix}{{\overset{\sim}{X}}_{r}^{(P)} = \{ {\begin{matrix}{{\sqrt{M}v_{m}1_{L}},} & {{r = m},} \\{0_{1 \times L},} & {r \neq m}\end{matrix},} } & (87)\end{matrix}$where ν_(m) is user s signal sent by transmit antenna m=1, . . . , Mduring the pilot transmission needed for estimating h_(n,m). Hence, userd received signal at antenna n during pilot transmission can be given by

$\begin{matrix}\begin{matrix}{{\overset{\sim}{Y}}_{n}^{(P)} = {{\sqrt{\rho_{\tau}}F_{L}h_{n,m}v_{m}} + {\overset{\sim}{W}}_{n}^{(P)}}} \\{= {{\sqrt{\rho_{\tau}}F_{L}g_{n,m}} + {\overset{\sim}{W}}_{n}^{(P)}}}\end{matrix} & (88)\end{matrix}$where g_(n,m)=h_(n,m)ν_(m) is the user d equivalent channel impulseresponse between receive antenna n and transmit antenna m. Using MMSE,channel estimation for the equivalent channel g_(n,m) isĝ _(n,m)=√{square root over (ρ_(τ))}F _(L) ^(H)(ρ_(τ) F _(L) F _(L) ^(H)+I _(L))⁻¹ {tilde over (Y)} _(n) ^((P)),  (89)and hence,Σ_(ĝ) =

{ĝ _(n,m) ĝ _(n,m) ^(H)}=Σ_(ĥ),Σ_({tilde over (g)}) =

{{tilde over (g)} _(n,m) {tilde over (g)} _(n,m)^(H)}=Σ_({tilde over (h)}),  (90)where ˜g_(n,m)=g_(n,m)−ĝ_(n,m) is the user d equivalent channelestimation error, and Σ_(ĥ), and Σ_({tilde over (h)}) are defined in Eq.(82), and Eq. (83), respectively. Therefore the user d estimatedequivalent channel, and the estimation error at subcarrier k are

$\begin{matrix}{{{\hat{G}}_{k} = {F_{k}\hat{g}}},\mspace{14mu}{\hat{g} = \begin{bmatrix}{\hat{g}}_{1,1} & \ldots & {\hat{g}}_{1,M} \\\vdots & \ddots & \vdots \\{\hat{g}}_{N_{d},1} & \ldots & {\hat{g}}_{N_{d},M}\end{bmatrix}},{{\overset{\sim}{G}}_{k} = {F_{k}\overset{\sim}{g}}},\mspace{14mu}{\overset{\sim}{g} = {\begin{bmatrix}{\overset{\sim}{g}}_{1,1} & \ldots & {\overset{\sim}{g}}_{1,M} \\\vdots & \ddots & \vdots \\{\overset{\sim}{g}}_{N_{d},1} & \ldots & {\overset{\sim}{g}}_{N_{d},M}\end{bmatrix}.}}} & (91) \\{{Futhermore},{\sigma_{\hat{G},k}^{2} = {{\frac{1}{N_{d}M}{Tr}\{ {\{ {F_{k}\hat{g}{\hat{g}}^{H}F_{k}^{H}} \}} \}} = \sigma_{\hat{H},k}^{2}}},{\sigma_{\overset{\sim}{G},k}^{2} = {{\frac{1}{N_{d}M}{Tr}\{ {\{ {F_{k}\overset{\sim}{g}{\overset{\sim}{g}}^{H}F_{k}^{H}} \}} \}} = \sigma_{\overset{\sim}{H},k}^{2}}},} & (92)\end{matrix}$where σ_(Ĥ,k) ², and σ_({tilde over (H)},k) ² are defined by Eq. (85).Thus user d can achieve the rate

$\begin{matrix}{R_{d} \geq {{\frac{T_{d} - M}{T_{d}K_{d}}{\sum\limits_{k = 1}^{L}\;{\{ {{\log\;\det\; I_{N_{d}}} + {\rho_{H,k}\frac{{\overset{\_}{G}}_{k}{\overset{\_}{G}}_{k}^{H}}{M}}} \}}}} + {\frac{1}{K_{d}}{\sum\limits_{k = {L_{d} + 1}}^{K_{d}}\;{\{ {{\log\;\det\; I_{N_{d}}} + {\rho_{H,k}\frac{{\overset{\_}{G}}_{k}{\overset{\_}{G}}_{k}^{H}}{M}}} \}.}}}}} & (93)\end{matrix}$

Furthermore, the user s received signal at the subcarriers with pilotsk=1, . . . , L, isZ _(k)=√{square root over (ρ_(τ))}D _(k) v+Ξ _(k),  (94)where v=[ν₁ ^(H), . . . , ν_(M) ^(H)]^(H) is the vector containing thestatic user signal. Define, the received signal at the L subcarrierscontaining pilots to be

$\begin{matrix}{{\overset{\_}{\overset{\_}{Z}} = {{\sqrt{\rho_{\tau}}{\overset{\_}{\overset{\_}{D}}}_{v}} + \overset{\_}{\overset{\_}{\Xi}}}},{where}} & (95) \\{{\overset{\_}{\overset{\_}{Z}} = \begin{bmatrix}Z_{1} \\\vdots \\Z_{L}\end{bmatrix}},{\overset{\_}{\overset{\_}{D}} = \begin{bmatrix}D_{1} \\\vdots \\D_{L}\end{bmatrix}},{\overset{\_}{\overset{\_}{\Xi}} = {\begin{bmatrix}\Xi_{1} \\\vdots \\\Xi_{L}\end{bmatrix}.}}} & (96)\end{matrix}$Thus user s can achieve the rate

$\begin{matrix}{{R_{s} \geq {\frac{1}{T_{d}K_{d}}\{ {{\log\;\det\; I_{N_{s}}} + {\frac{1}{\{ \lambda_{i}^{- 2} \}}\overset{\_}{\overset{\_}{D}}\mspace{11mu}{\overset{\_}{\overset{\_}{D}}}^{H}}} \}}},} & (97)\end{matrix}$which represents the “for free” achievable rate using productsuperposition transmission.

IV. Disparity in Coherence Bandwidth

In this section, the downlink transmission for two users with disparityin bandwidth, i.e. K_(d)≠K_(s), but equality in time, i.e., T_(s)=T_(d),is examined. The channels of user s and user d stay constant during thesame coherence time T_(s)=T_(d)=T. The two users are assumed to have thesame selectivity over time, and hence, the value of T could be short orlong. Consider the case when L_(d)=1, i.e., the channel of user d isflat fading (H_(k)=H, ∀k), whereas L_(s)≥1, i.e., users channel isfrequency selective. The system can be given achievable rates obtainedby applying the product superposition transmission and the gain over theconventional transmission.

For this system, the product superposition transmission is as follows.Since the channel of user d is the same for all the subcarriers, thechannel can be estimated at one subcarrier, k₁, every T time slots. Onthe other hand, channel estimation at all the subcarriers is needed foruser S. This section shows the gain provided by product superpositiontransmission when channel estimation is done in the frequency domainwithout interpolation. For frequency-domain channel estimation withinterpolation, and time-domain channel estimation, product superpositiontransmission still can provide gain over conventional transmission, asshown in Section III-D. The analysis of the latter two cases is removedhere for brevity. Therefore at subcarrier k₁, the pilot signals can besent so that the two users can estimate their channels, D_(k) ₁ , and H,and furthermore, at the other subcarriers k≠k₁, product superpositioncan be used to provide for free rate for user d. Hence, at thesubcarriers k≠k₁, the transmitted signal isX _(k)=[√{square root over (M)}U _(k) ,U _(k) V _(k)],  (98)where U_(k)∈

^(M×M) is the signal of user d, and V_(k)∈

^(M×T−M). Hence, the users received signal at the first M time slots isZ _(k) ^((p))=√{square root over (ρ_(τ,k))}D _(k) U _(k)+Ξ_(k)^((p))=√{square root over (ρ_(τ,k))}J _(k)+Ξ_(k) ^((p))  (99)where J_(k)=D_(k)U_(k) is the user s equivalent channel matrix atsubcarrier k. Following the same analysis in Section III, the MMSEestimate of the channel is

$\begin{matrix}{{{\hat{J}}_{k} = {{\gamma_{\tau,k}J_{k}} + {\frac{\gamma_{\tau,k}}{\sqrt{\rho_{\tau,k}}}\Xi_{k}^{(p)}}}},} & (100) \\{{\sigma_{\hat{J}}^{2} = {\gamma_{\tau,k}{f_{k}}^{2}}},} & (101)\end{matrix}$and furthermore, the user s channel estimation error is

$\begin{matrix}{{\overset{\sim}{J}}_{k} = {{J_{k} - {\hat{J}}_{k}} = {{( {1 - \gamma_{\tau,k}} )J_{k}} - {\frac{\gamma_{\tau,k}}{\sqrt{\rho_{\tau,k}}}{\Xi_{k}^{(p)}.}}}}} & (102)\end{matrix}$Therefore, the normalized MMSE of the user s estimated channel is

$\begin{matrix}{\sigma_{\overset{\sim}{J}}^{2} = {\frac{\gamma_{\tau,k}}{\rho_{\tau,k}}.}} & (103)\end{matrix}$Therefore, using K_(s) subcarriers and during T time slots, user s canachieve the rate

$\begin{matrix}{{R_{s} \geq {\frac{T - M}{{TK}_{s}}{\sum\limits_{k = 1}^{K_{s}}\;{\{ {{\log\;\det\; I_{N_{s}}} + {\rho_{J,k}\frac{{\overset{\_}{J}}_{k}{\overset{\_}{J}}_{k}^{H}}{M}}} \}}}}},} & (104)\end{matrix}$where J _(k)=Ĵ_(k)/σ_(ĵ) _(k) , and

$\begin{matrix}{\rho_{J,k} = {\frac{\rho_{d,k}\sigma_{\hat{J}}^{2}}{1 + {\rho_{d,k}\sigma_{\overset{\sim}{J}}^{2}}}.}} & (105)\end{matrix}$

Furthermore, using K_(s) subcarriers and during T time slots, user d canachieve the rates

$\begin{matrix}{{{R_{d} \geq {\frac{M}{{TK}_{s}}{\sum\limits_{{k = 1},{k \neq k_{1}}}^{K_{s}}\;{\{ {{\log\;\det\; I_{N_{d}}} + {\rho_{H,k}\frac{\overset{\_}{H}{\overset{\_}{H}}^{H}}{M}}} \}}}}},{where}}{{\rho_{H,k} = \frac{\sigma_{\hat{H},k}^{2}}{{\{ \epsilon_{i}^{- 2} \}} + \sigma_{\overset{\sim}{H},k}^{2}}},}} & (106)\end{matrix}$and ∈_(i) ⁻² i is any unordered eigenvalue of {tilde over (V)}_(k){tildeover (V)}_(k) ^(H), {tilde over (V)}_(k)=[ρ_(r)I, ρ_(δ)V_(k)V_(k) ^(H)].Hence, the user d achievable rate in the above equation demonstrates the“for free” rate obtained by using product superposition transmission.

V. Disparity in Both Coherence Time and Coherence Bandwidth

In this section, consider the scenario when the users have disparity inboth coherence time and coherence bandwidth at the same time, i.e.,T_(d)≠T_(s) and L_(d)≠L_(s). In the case when L_(d)=1, L_(s)≥1, andT_(s)=qT_(d), where q=1, 2, . . . , the coherence time of users is amultiple integer of user d. Two versions of product superpositiontransmissions can be used to obtain “for free” gain over theconventional transmission. The first one is product superpositiontransmission with respect to user d where user s obtains the “for free”rate, and the second is product superposition transmission with respectto user s where user d obtains the “for free” gain.

A. Product Superposition Transmission with Respect to User d

Since T_(s)=qT_(d), the interval T_(s) can be divided into qsubintervals. During the first subinterval of length T_(d) slots, thetransmitter can send pilots during the first M resource elements atsubcarrier k₁ so that user d can estimate its channel H, and furthermoreuser s can estimate its channel at subcarrier k₁. Hence,

$\begin{matrix}{{\hat{H} = {{\frac{\rho_{\tau,k_{1}}}{\rho_{\tau,k} + 1}H} + {\frac{\sqrt{\rho_{\tau,k_{1}}}}{\rho_{\tau,k_{1}} + 1}W_{k_{1}}^{(p)}}}},{{\hat{D}}_{k_{1}} = {{\gamma_{\tau,k_{1}}D_{k_{1}}} + {\frac{\gamma_{\tau,k_{1}}}{\sqrt{\rho_{\tau,k_{1}}}}{\Xi_{k_{1}}^{(p)}.}}}}} & (107)\end{matrix}$During the following (q−1) subintervals, user d utilizes pilots toestimate its channel, whereas, user s already has estimated its channelduring the first subinterval. Hence, product superposition transmissioncan be used at subcarrier k₁ to obtain the “for free” rate for user s.Therefore, the transmitted signal at the subcarrier k₁ during the firstM resource elements isX _(k) ₁ ^((p)) =√{square root over (M)}V,  (108)where V is the user s signal. Hence, the user d received signal duringthe first M resource elements isY _(k) ₁ ^((p))=√{square root over (ρ_(τ,k) ₁ )}G+W _(k) ₁^((p)),  (109)where G=HV is the user d equivalent channel matrix. Therefore, the MMSEestimate of the channel isY _(k) ₁ ^((p))=√{square root over (ρ_(τ,k) ₁ )}G+W _(k) ₁^((p)),  (110)and furthermore,

$\begin{matrix}{{\sigma_{\hat{G}}^{2} = \frac{\rho_{\tau,k_{1}}}{1 + \rho_{\tau,k_{1}}}},} & (111)\end{matrix}$Hence, the user d channel estimation error is

$\begin{matrix}{\overset{\sim}{G} = {{G - \hat{G}} = {{\frac{1}{1 + \rho_{\tau,k_{1}}}G} - {\frac{\sqrt{\rho_{\tau,k_{1}}}}{1 + \rho_{\tau,k_{1}}}{W_{k_{1}}^{(p)}.}}}}} & (112)\end{matrix}$and furthermore,

$\begin{matrix}{\sigma_{\overset{\sim}{G}}^{2} = {\frac{1}{1 + \rho_{\tau,k_{1}}}.}} & (113)\end{matrix}$Hence, user d can achieve the rate

$\begin{matrix}{R_{d} \geq {{\frac{1}{q}( {{\frac{T_{d} - M}{T_{d}K_{d}}\{ {{\log\;\det\; I_{N_{d}}} + {\rho_{H,k_{1}}\frac{\overset{\_}{H}{\overset{\_}{H}}^{H}}{M}}} \}} + {\frac{1}{K_{d}}{\sum\limits_{{k = 1},{k \neq k_{1}}}^{K_{d}}\;{\{ {{\log\;\det\; I_{N_{d}}} + {\rho_{H,k}\frac{\overset{\_}{H}{\overset{\_}{H}}^{H}}{M}}} \}}}}} )} + {( {1 - \frac{1}{q}} ){( {{\frac{T_{d} - M}{T_{d}K_{d}}\{ {{\log\;\det\; I_{N_{d}}} + {\rho_{G,k_{1}}\frac{\overset{\_}{G}{\overset{\_}{G}}^{H}}{M}}} \}} + {\frac{1}{K_{d}}{\sum\limits_{{k = 1},{k \neq k_{1}}}^{K_{d}}\;{\{ {{\log\;\det\; I_{N_{d}}} + {\rho_{G,k}\frac{\overset{\_}{G}{\overset{\_}{G}}^{H}}{M}}} \}}}}} ).}}}} & (114)\end{matrix}$Furthermore, the achievable rate by user s is

$\begin{matrix}{{R_{s} \geq {( {1 - \frac{1}{q}} )\frac{M}{T_{d}K_{d}}\{ {{\log\;\det\; I_{N_{s}}} + {\rho_{H,k_{1}}\frac{{\overset{\_}{D}}_{k_{1}}{\overset{\_}{D}}_{k_{1}}^{H}}{M}}} \}}},} & (115)\end{matrix}$which demonstrates the “for free” rates obtained by productsuperposition transmission over conventional transmission.

B. Product Superposition with Respect to User s

The second product superposition scheme makes use of the disparity incoherence bandwidth between the two users. Hence, the productsuperposition with respect to user s is the same as the one given inSection IV yielding the user achievable rates that are given in Eq.(104), and Eq. (106), where the latter is the “for free” rate over theconventional transmission.

VI. Numerical Results

The numerical results were averaged over 10000 Monte Carlo runs. In eachrun, the channel coefficients and the users noises are independent andrandomly generated. The system sum-rate denotes, for the case ofconventional transmission, the interference-free rate of one user, andfor the case of product superposition transmission, the summation of theinterference-free rate and the “for free” rate of the two users. For thesystem with disparity in coherence time discussed in Section III,consider a system of M=4 transmitting antennas, N_(d)=4, and N_(s)=4receive antennas at user d, and user s, respectively. Furthermore, userd and user s channels have L_(s)=L_(d)=2 taps, and the number ofsubcarriers is K=12 subcarriers and the coherence time of user d isT_(d)=20 time slots, i.e. the size of the considered resource block is12 subcarriers and 20 time slots.

FIGS. 3 and 4 compare the conventional transmission, and the productsuperposition transmission when the users channels have disparity incoherence time. FIG. 3 shows the rates (sum-rate) versus the transmittedSNR when frequency-domain channel estimation is used withoutinterpolation. There is disparity in coherence time. Notice from FIG. 3,the gain obtained when product superposition transmission is used overthe conventional transmission. Furthermore, FIG. 4 shows that productsuperposition transmission can still provide gain when interpolation isused for frequency domain channel estimation with interpolation. Therates (sum-rate) versus the SNR are when there is disparity in coherencetime. In FIGS. 3 and 4, the curves with no power optimization denote thecase when ρ_(τ,k)=ρ_(d,k)=ρ, whereas the curves with power optimizationdenote the case when ρ_(τ,k), ρ_(d,k) are given in Eq. (31).

For the same system parameters, and interpolation is used to estimateuser d channel, FIGS. 5 and 6 demonstrate the effect of using an optimaluser s interpolation matrix B_(k) given in Eq. (76) on the productsuperposition gain. In FIG. 5, the product superposition transmission isshown when B_(k) is optimal, linear, or random. Frequency-domain channelestimation with interpolation is used. Furthermore, FIG. 6 compares theMMSE of channel estimation for conventional transmission, and productsuperposition transmission with linear, and optimal B_(k). There isdisparity in coherence time, and frequency-domain channel estimationwith interpolation is used.

For simulating the system with time-domain channel estimation, considerM=4 transmitting antennas, N_(d)=2, and N_(s)=4 receive antennas at userd, and users, respectively. Furthermore, user d and user s channels canhave L_(s)=L_(d)=5 taps, and the number of subcarriers considered isK=10 subcarriers and the coherence time of user d is T_(d)=8 time slots.FIG. 7 shows the product superposition gain compared to the conventionaltransmission. The sum-rate versus SNR is when there is disparity incoherence time, and time-domain channel estimation is used.

FIG. 8 shows the product superposition gain over the conventional systemwhen the channels of user d and user s have disparity in coherencebandwidth. The sum-rate versus SNR when there is disparity in coherencebandwidth. Consider a system of M=4 transmitting antennas, N_(d)=2, andN_(s)=2 receive antennas at user d, and users, respectively.Furthermore, user d and user s channels have the same coherence timeT_(s)=T_(d)=T=8 time slots. Furthermore, user d has L_(d)=1 channel tap,whereas, the channel of user s has L_(s)=2 taps, and the number ofsubcarriers considered is K=8 subcarriers.

FIGS. 9 and 10 show the sum-rate increase when product superpositiontransmission is used for the case of disparity in both coherence timeand coherence bandwidth. Consider a system with M=4 transmittingantennas, N_(d)=2, and N_(s)=2 receive antennas. The disparity in bothcoherence time and coherence bandwidth is simulated when the channels ofuser d and users have L_(d)=1, and L_(s)=3 taps, respectively, andfurthermore, the coherence time of user d is T_(d)=8 time slots, andthat of users is T_(s)=24 time slots, i.e. q=3. In FIG. 9, the twoproduct superposition transmissions: with respect to user d, and withrespect to user s are compared their corresponding conventionaltransmissions. The sum-rate versus SNR is when there is disparity inboth coherence time and coherence bandwidth. FIG. 10 shows the increasein the system rate region obtained via product superposition schemecompared to the conventional transmission rate region obtained by TDMAbetween the two users. The system rate region is for a system withdisparity in both coherence time and coherence bandwidth.

VII. Conclusion

The first part of this disclosure has presented multiuser wirelesschannels when the users have disparity in fading conditions, inparticular disparity in coherence time and coherence bandwidth. A newsource of gain, called coherence diversity, can be obtained by makinguse of this disparity phenomenon. OFDM downlink transmission for twousers was considered in three different scenarios of disparity: incoherence time, in coherence bandwidth, and in both coherence time andcoherence bandwidth. In each scenario a version of product superpositiontransmission was used to obtain the gains in the achievable rates.Numerical simulations were also provided to calculate the gains in thesystem achievable rates compared to conventional transmissiondemonstrating the superiority of the proposed schemes.

Coherence Diversity for Multiple Users I. Introduction

In a wireless network, variations in node mobility and scatteringenvironment may easily produce unequal link coherence times. But theperformance limits of wireless networks under unequal link coherencetimes have been for the most part an open problem.

Even under identical coherence times, understanding the performancelimits of many wireless networks under block fading or related modelshas been far from trivial, with some key results under identical fadingintervals being realized only very recently. For a two-usermultiple-input single-output (MISO) broad-cast channel with receive-sidechannel state information (CSIR) and finite precision transmit-sidechannel state information (CSIT), the degrees of freedom may collapse tounity under (non-singular) correlated fading. A broadcast channel withheterogeneous CSIT, i.e., the CSIT with respect to different links maybe perfect, delayed, or non-existent, has been considered. In this case,a collapse of degrees of freedom for a two-receiver broadcast may occuras long as CSIT with respect to one link is missing. These were settledin the positive using the idea of aligned image sets. Furthermore, anouter bound for a K-receiver MISO broadcast channel was developed wherethere is CSIT with respect to some link gains and delayed CSIT withrespect to other link gains. For the 3-receiver case, when thetransmitter has CSIT for one receiver and delayed CSIT for the othertwo, a transmission scheme achieving 5/3 sum degrees of freedom can befound. For the same system, a transmission scheme was proposed that canachieve 9/5 sum degrees of freedom. A broadcast channel with delayedCSIT was demonstrated showing that even completely outdated channelfeedback is still useful. A scenario of mixed CSIT (imperfectinstantaneous and perfect delayed) can also be considered.

Considering a two-receiver broadcast channel with CSIR but no CSIT underi.i.d. fast fading, it can be shown that TDMA is degrees of freedomoptimal. This result can be extended to multiple receivers and to awider class of fading distributions and fading dynamics (not includingblock fading). This can be based on the notion of stochastic equivalenceof links with respect to the transmitter.

For the broadcast channel, a summary of the results of the second partof this disclosure is as follows. Begin by settling the open problem ofthe degrees of freedom of the multi-receiver block-fading broadcastchannel with identical fading intervals. We show that with CSIR but noCSIT, the degrees of freedom is limited to time-sharing. In the absenceof CSIR (and CSIT) it can be shown that once again the degrees offreedom cannot be improved beyond time sharing.

Now consider unequal fading intervals, where the perspective for theavailability and cost of channel state information is quite distinctfrom the case of equal fading intervals. Specifically, the normalizedper-transmission cost of acquiring CSIR, e.g. via pilots, is closelyrelated to the block length, therefore the normalized cost of CSIR forlinks with un-equal coherence times may vary widely. It follows thatwhen fading intervals are unequal, any assumption of free CSIR mayobscure important features of the problem. Therefore consider a modelwithout free availability of CSIR. i.e., one where the cost of CSIR isaccounted for. For achievable degrees of freedom of the multi-receiverbroadcast channel, a generalization of the method of productsuperposition to multiple receivers with coherence times of arbitraryinteger ratios may be used, and without free CSIR. Also, availability ofCSIT is not assumed. For example, this method may be considered for thespecial case of a two-receiver broadcast channel were one receiver has avery long coherence time compared with the other. This achievable ratecan be obtained by transmitting a pilot whenever one or more receiversexperience a fading transition, and then during each pilot transmissionexactly one (other) receiver who does not need the pilot cansimultaneously utilize the channel for data transmission withoutcontaminating the pilot. This leads to degrees of freedom gains that aredirectly tied to the disparity of coherence times, and are thereforecalled coherence diversity.

If the coherence time is at least twice the number of transmit andreceive antennas, the obtained degrees of freedom can be shown to meetthe upper bound in four cases: When the number of transmit antennas isless than or equal to the number of antennas at every receiver, when allthe receivers have the same number of antennas, when the coherence timesof the receivers are very long compared to one receiver, or when all thereceivers have identical coherence times. The development of outerbounds for this problem makes use of the idea of channel enhancement,which comprises increasing the coherence time of all receivers to matchthe coherence time of the slowest channel.

The inner bounds for coherence diversity can be further extended to thecase of multiple receivers experiencing fading block lengths ofarbitrary ratio or alignment. Unaligned block fading intervals bring tomind blind interference alignment. Consider a version of bindinterference alignment takes into account the full cost of CSIR viatraining; in that framework the synergies between blind interferencealignment and product superposition are examined.

For the block-fading multiple-access channel, the capacity in theabsence of CSIR is unknown. In fact, the capacity of a point-to-pointchannel under this condition is also unknown except in certain specialcases. In the single-input multiple-output (SIMO) block-fading multipleaccess without CSIR, the sum capacity can be achieved by activating nomore than T receivers. Also, for a two-receiver single-inputsingle-output (SISO) multiple access channel with i.i.d. fast fading(all the receivers have coherence time of length 1), a non-naivetime-sharing inner bound and a cooperative outer bound on the capacityregion can be provided. Furthermore, a multi-receiver multiple accesschannel with identical coherence times where the receivers are equippedwith single antenna where an inner bound on the network sum capacity canbe provided based on successive decoding, and an outer bound can beobtained based on assuming cooperation between the transmitters.

The results for the multiple-access channel are as follows: begin byhighlighting bounds on the degrees of freedom of the block-fading MIMOmultiple access channel with identical coherence times in the absence offree CSIR. A conventional pilot-based scheme emitting individual andseparate pilots from (e.g., a subset of) the antennas of receivers isconsidered that subsequently allows the receiver to performzero-forcing. This method can be shown to partially meet the cooperativeouter bound. In particular, this method can always achieves the optimalsum degrees of freedom, and in some cases can be optimal throughout thedegrees of freedom region. For the case of unequal coherence times, thesame transmission technique can be employed with pilots transmitted atthe fading transition times of every active receiver. The outer boundcan once again build on the concept of enhancing the channel byincreasing the receivers coherence times so that the receivers of theenhanced channel have identical coherence times.

The key results of the paper are summarized in the table of FIG. 11 forbroadcast channel and in the table of FIG. 12 for multiple accesschannel, where

${N_{i}^{*} = {\min\{ {M,N_{i},{\frac{T_{i}}{2}}} \}}},{N_{\max} = {\max_{j \in {\mathbb{J}}}{\{ N_{j} \}\mspace{14mu}{and}\mspace{14mu} j_{\min}}}}$is the receiver with the shortest coherence time in

.

II. Broadcast Channel with Identical Coherence Times

Consider a K-receiver MIMO broadcast channel where the transmitter isequipped with M antennas and receiver k is equipped with N_(k) antennas,k=1, . . . , K. The received signal at receiver k isy _(k)(n)= H _(k)(n)x(n)+z _(k)(n),k=1, . . . ,K,  (116)where x(n)∈

^(M×1) is the transmitted signal, z_(k)(n)∈

^(N) ^(k) ^(×1) is receiver k i.i.d. Gaussian additive noise and H_(k)(n)∈

^(N) ^(k) ^(×M) is the receiver k Rayleigh block-fading channel matrixwith coherence interval of length T_(k) time slots, at the discrete timeindex n. One time slot is equivalent to a single transmission symbolperiod, and all T_(k) are positive integers. Assume no CSIT, meaning therealization of H _(k)(n) is not known at the transmitter, whereas itsdistribution (including the length of the coherence time, and itstransition) is globally known at the transmitter and at all receivers.

Also assume that there are K independent messages associated with ratesR₁(ρ), . . . , R_(K)(ρ) to be communicated from the transmitter to the Kreceivers at ρ signal-to-noise ratio. The degrees of freedom at receiverk achieving rate R_(k)(ρ) can be defined as

$\begin{matrix}{d_{k} = {\lim\limits_{\rhoarrow\infty}{\frac{R_{k}(\rho)}{\log(\rho)}.}}} & (117)\end{matrix}$The degrees of freedom region of a K-receiver MIMO broadcast is definedas

= { ( d 1 , … ⁢ , d K ) ∈ + K | ∃ ( R 1 ⁡ ( ρ ) , … ⁢ , R K ⁡ ( ρ ) ) ∈ C ⁡ (ρ ) , ⁢ d k = lim ρ → ∝ ⁢ R k ⁡ ( ρ ) log ⁡ ( ρ ) , k ∈ 1 , … ⁢ , K } , ( 118)where C(ρ) is the capacity region at ρ signal-to noise ratio.

Assume that the receivers have identical coherence times, where thecoherence times are perfectly aligned, and furthermore, have the samelength, namely T. For the capacity to be determined, it is sufficient tostudy the capacity of only one coherence time. Define Y_(k)(n)∈

^(N) ^(k) ^(×T), X(n)∈

^(M×T) to be the received signal at receiver k=1, . . . , K and thetransmitted signal, respectively, during the coherence time T,Y _(k) =H _(k) X _(k) +Z _(k) ,k=1, . . . ,K,  (119)where H_(k)(n)∈

^(N) ^(k) ^(×M) is the receiver k channel matrix which remains constantduring the interval T.

When there is CSIR, the degrees of freedom optimality of TDMA for tworeceivers with T=1 can be determined. Furthermore, the result can beextended to an arbitrary number of receivers and for a wider class offading distribution. Since there is no CSIT, and furthermore, thereceivers have identical coherence times, namely T, the receivers arestochastically equivalent (indistinguishable) with respect to thetransmitter. As a result, TDMA is enough to achieve the degrees offreedom region of the system, i.e. the degrees of freedom region can begiven by,

= { ( d 1 , … ⁢ , d K ) ∈ + K | ∑ i = 1 K ⁢ ⁢ d i min ⁢ { M , N i } ≤ 1 } .As will be further discussed in, e.g., paragraph 0145, this result forT≥1 can be extended to show that TDMA is degrees of freedom optimal.

Now assume that, for a K-receiver broadcast channel, there is no CSIR.As long as the receivers have identical coherence times, the receiversare still stochastically equivalent. In the sequel, it can be shows thatTDMA is enough to achieve the degrees of freedom region of the system.

Theorem 1:

Consider a K-receiver broadcast channel with identical coherence timesT. When there is no CSIT or CSIR meaning that the channel realization isnot known, but the channel distribution is globally known, the degreesof freedom region of the channel is given by,

= { ( d 1 , … ⁢ , d K ) ∈ + K | ∑ i = 1 K ⁢ ⁢ d i N i * ⁡ ( 1 - N i * T ) ≤1 } . ⁢ where ⁢ ⁢ N i * = min ⁢ { M , N i ,  T i 2  } . ( 120 )

Proof:

A simple time division multiplexing between the receivers achieves thedegrees of freedom region. The remainder of the proof is dedicated tofinding a corresponding outer bound. Without loss of generality, assumeN₁≤ . . . ≤N_(K). When M≤N₁, the cooperative outer bound for the sumdegrees of freedom is

= { ( d 1 , … ⁢ , d K ) ∈ + K | ∑ i = 1 K ⁢ ⁢ d i N i * ⁡ ( 1 - N i * T ) ≤1 } . ( 121 )which is tight against the TDMA inner bound. When M≥N₁, to obtain theouter bound the following Lemma is introduced.

Lemma 1:

For the above MIMO K-receiver broadcast channel, define Y=[Y₁ ^(H), Y₂^(H), . . . , Y_(K) ^(H)]^(H) to be the matrix that contains allreceived signals during T interval, and Y _(j)∈

^(1×T) is row j of Y and {tilde over (Y)}_(S) is the matrix constructedfrom excluding the set S of the rows from the matrix Y. Then,I(X;Y _(j) |U,{tilde over (Y)} _({j,l}))=I(X;Y _(l) |U,{tilde over (Y)}_({j,l})),  (122)and furthermore,I(X;Y _(j) |U,{tilde over (Y)} _({j,l}))≥I(X;Y _(j) |U,{tilde over (Y)}_({j,l}) ,Y _(l)),  (123)where Uset→X→Y forms a Markov Chain. The proof of Lemma 1 is providedbelow.

Now, the outer bound for the case when M≥N₁ can be found. Since thereceivers have the same noise variance, the system is considereddegraded,R _(k) ≤I(U _(k) ;Y _(k) |U ^(k−1)),k≠K,R _(K) ≤I(X;Y _(K) |U ^(K−1)),  (124)where U₁→ . . . →U_(K−1)→X→(Y₁, . . . , Y_(K)) forms a Markov Chain, andU₀ is a trivial random variable. Using the chain rule, Eq. (124) can bewritten asR _(k) ≤I(X;Y _(k) |U ^(k−1))−I(X;Y _(k) |U ^(k)),k≠K,R _(K) ≤I(X;Y _(K) |U ^(K−1)).  (124)Define r_(k) to be the degrees of freedom of the term I(X; Y_(k)|U^(k)),where

$0 \leq r_{k} \leq {{N_{k}^{*}( {1 - \frac{M}{T}} )}.}$Furthermore, the degrees of freedom of I(X; Y₁) is bounded by the singlereceiver bound, i.e.

${N_{1}( {1 - \frac{M}{T}} )},$hence,

$\begin{matrix}{\mspace{79mu}{{{R_{1} \leq {{( {{N_{1}( {1 - \frac{M}{T}} )} - r_{1}} ){\log(\rho)}} + {o( {\log(\rho)} )}}},\mspace{20mu}{R_{k} \leq {{I( {X; Y_{k} \middle| U^{k - 1} } )} - {r_{k}{\log(\rho)}} + {o( {\log(\rho)} )}}},{k \neq 1},K}\mspace{20mu}{{R_{k} \leq {{I( {X; Y_{k} \middle| U^{K - 1} } )}.\mspace{20mu}{Furthermore}}},}}} & (125) \\\begin{matrix}{{{r_{k}{\log(\rho)}} + {o( {\log(\rho)} )}} = {I( {X; Y_{k} \middle| U^{k} } )}} \\{\overset{(a)}{=}{{I( {X; Y_{k,{1:N_{k}^{*}}} \middle| U^{k} } )} +}} \\{{I( {{X; Y_{k,{{N_{k}^{*} + 1}:N_{k}}} \middle| U^{k} },Y_{k,{1:N_{k}^{*}}}} )} + {o( {\log(\rho)} )}} \\{\overset{(b)}{\geq}{{I( {X; Y_{k,{1:N_{k}^{*}}} \middle| U^{k} } )} + {o( {\log(\rho)} )}}} \\{\overset{(c)}{=}{{\sum\limits_{i = 1}^{N_{k}^{*}}\;{I( {{X; Y_{k,i} \middle| U^{k} },Y_{k,{{i + 1}:N_{k}^{*}}}} )}} + {o( {\log(\rho)} )}}} \\{\overset{(d)}{=}{{\sum\limits_{i = 1}^{N_{k}^{*}}\;{I( {{X; Y_{k,1} \middle| U^{k} },Y_{k,{{i + 1}:N_{k}^{*}}}} )}} + {o( {\log(\rho)} )}}} \\{{\overset{(e)}{\geq}{{N_{k}^{*}{I( {{X; Y_{k,1} \middle| U^{k} },Y_{k,{2:N_{k}^{*}}}} )}} + {o( {\log(\rho)} )}}},}\end{matrix} & (126)\end{matrix}$where Y_(k,i:j) denotes the matrix constructed from the rows i:j of thematrix Y_(k). (a), and (c) follow from the chain rule, and (b) followssince mutual information is non-negative. Furthermore, (d) follows fromLemma 1 and (e) follows since removing conditioning does not reduce theentropy. Therefore,

$\begin{matrix}{\mspace{79mu}{{{I( {{X; Y_{k,1} \middle| U^{k} },Y_{k,{2:N_{k}^{*}}}} )} \leq {{\frac{r_{k}}{N_{k}^{*}}{\log(\rho)}} + {{o( {\log(\rho)} )}.\mspace{20mu}{Furthermore}}}},}} & (127) \\\begin{matrix}{{I( {X; Y_{k} \middle| U^{k - 1} } )}\overset{(a)}{=}{{I( {X; Y_{k,{1:N_{k}^{*}}} \middle| U^{k - 1} } )} +}} \\{I( {{X; Y_{k,{{N_{k}^{*} + 1}:N_{k}}} \middle| U^{k - 1} },Y_{k,{1:N_{k}^{*}}}} )} \\{\overset{(b)}{=}{{I( {X; Y_{k,{1:N_{k - 1}^{*}}} \middle| U^{k - 1} } )} +}} \\{{I( {{X; Y_{k,{{N_{k - 1}^{*} + 1}:N_{k}^{*}}} \middle| U^{k - 1} },Y_{k,{1:N_{k - 1}^{*}}}} )} + {o( {\log(\rho)} )}} \\{\overset{(c)}{=}{{I( {X; Y_{{k - 1},{1:N_{k - 1}^{*}}} \middle| U^{k - 1} } )} +}} \\{{I( {{X; Y_{k,{{N_{k - 1}^{*} + 1}:N_{k}^{*}}} \middle| U^{k - 1} },Y_{{k - 1},{1:N_{k - 1}^{*}}}} )} + {o( {\log(\rho)} )}} \\{= {{r_{k - 1}{\log(\rho)}} +}} \\{{\sum\limits_{i = {N_{k - 1}^{*} + 1}}^{N_{k}^{*}}\;{I\begin{pmatrix}{{X; Y_{k,i} \middle| U^{k - 1} },} \\{Y_{{k - 1},{1:N_{k - 1}^{*}}},Y_{k,{{i + 1}:N_{k}^{*}}}}\end{pmatrix}}} + {o( {\log(\rho)} )}} \\{\overset{(d)}{\leq}{{r_{k - 1}{\log(\rho)}} +}} \\{{( {N_{k}^{*} - N_{k - 1}^{*}} ){I\begin{pmatrix}{{X; Y_{{k - 1},1} \middle| U^{k - 1} },} \\Y_{{k - 1},{2:N_{k - 1}^{*}}}\end{pmatrix}}} + {o( {\log(\rho)} )}} \\{\overset{(e)}{\leq}{{r_{k - 1}{\log(\rho)}} + {( {N_{k}^{*} - N_{k - 1}^{*}} )\frac{r_{k - 1}}{N_{k - 1}^{*}}{\log(\rho)}} + {o( {\log(\rho)} )}}} \\{{\leq {{\frac{N_{k}^{*}}{N_{k - 1}^{*}}r_{k - 1}{\log(\rho)}} + {o( {\log(\rho)} )}}},}\end{matrix} & (128)\end{matrix}$here (a) and (b) follow from applying the chain rule, and I(X; Y_(k,N)_(k) _(*) _(+1:N) _(k) |U^(k−1),Y_(k,1:N) _(k) )=o(log(ρ)) since morereceive antennas than N_(k)* does not increase the degrees of freedom.Furthermore, (c) follows since Y_(k,1:N) _(k−1) * and Y_(k−1,1:N)_(k−1) * are statistically the same. (d) follows from applying Lemma 1and (e) follows from Eq. (127). Therefore,

$\begin{matrix}{\mspace{79mu}{{d_{1} \leq {{N_{1}( {1 - \frac{M}{T}} )} - r_{1}}},\mspace{20mu}{d_{k} \leq {{\frac{N_{k}^{*}}{N_{k - 1}^{*}}r_{k - 1}} - r_{k}}},{i \neq 1},K,\mspace{20mu}{d_{K} \leq {\frac{N_{K}^{*}}{N_{K - 1}^{*}}{r_{K - 1}.\mspace{20mu}{Hence}}}},}} & (129) \\{{{\sum\limits_{i = 1}^{K}\;\frac{d_{i}}{N_{i}^{*}( {1 - \frac{M}{T}} )}} \leq {1 + {\sum\limits_{i = 2}^{K}\;\frac{r_{k - 1}}{N_{k - 1}^{*}( {1 - \frac{M}{T}} )}} - {\sum\limits_{i = 1}^{K - 1}\;\frac{r_{k}}{N_{k}^{*}( {1 - \frac{M}{T}} )}}}},{= 1},} & (130)\end{matrix}$where the last inequality follows since the two summations on the righthand side cancel each other. Thus, the degrees of freedom region isbounded by TDMA of the single receiver points

${N_{k}^{*}( {1 - \frac{M}{T}} )},$which is maximized to be

${N_{k}^{*}( {1 - \frac{N_{k}^{*}}{T}} )},$completing the proof of Theorem 1.

III. Broadcast Channel with Heterogeneous Coherence Times

Consider the K-receiver broadcast channel defined in Eq. (116) wherethere is no CSIT or CSIR. Consider that the receivers have perfectlyaligned coherence times with integer ratio, i.e.,

${\frac{T_{k}}{T_{k - 1}} \in {\mathbb{Z}}},$∀k. FIG. 13 denotes three receivers where T₃=2T₂=4T₁. In this system,the receivers are no longer stochastically equivalent, and hence, TDMAinner bound is no longer tight.

In the following subsections, product superposition transmission inrevisited, and a product superposition transmission for the K-receiverbroadcast channel defined in Eq. (116) calculating the achievabledegrees of freedom region is given. An outer bound on the degrees offreedom region is also given and the tightness of these bounds is shown,and hence, the optimality of the achievable product superposition schemefor four cases. Numerical examples are also provided.

A. Product Superposition Scheme

A two-receiver broadcast channel with no CSIT and with mixed CSIR havebeen studied; one static receiver has very long coherence time, hence,there is CSIR for this receiver, and one dynamic receiver has shortcoherence time T_(d), hence, there is no CSIR for this receiver. It canbe shown that TDMA is suboptimal in such a broadcast channel and aproduct superposition scheme is disclosed as follows. ConsiderM≥N_(S)≥N_(d), where N_(s), N_(d) are the number of receive antennas ofthe static and dynamic receivers, respectively. The transmitted signalisX=X _(s) X _(d),  (131)where X_(s)∈

^(M×N) ^(d) is the data matrix for the static receiver with i.i.d.

(0,1) elements, and X_(d)∈

^(N) ^(d) ^(×T) ^(d) is the signal matrix for the dynamic receiver,whereX _(d)=[I _(N) _(d) ,X _(δ)],  (132)I_(N) _(d) is an N_(d)×N_(d) identity matrix, and X_(δ)∈

^(N) ^(d) ^(×(T) ^(d) ^(−N) ^(d) ⁾ is the dynamic receiver data matrixhaving i.i.d

(0,1) elements. Therefore the signal received at the dynamic receiver,during T_(d) slots, is

$\begin{matrix}\begin{matrix}{Y_{d} = {{H_{d}{X_{s}\lbrack {I_{N_{d}},X_{\delta}} \rbrack}} + Z_{d}}} \\{{= {\lbrack {{\overset{\_}{H}}_{d},{{\overset{\_}{H}}_{d}X_{\delta}}} \rbrack + Z_{d}}},}\end{matrix} & (133)\end{matrix}$where H _(d)=H_(d)X_(s), and H_(d)∈

^(N) ^(d) ^(×M) is the dynamic receiver channel. The dynamic receiverestimates the equivalent channel H _(d) during the first N_(d) slots andthen decodes X_(δ) coherently based on the channel estimation. On theother hand, the received signal at the static receiver during the firstN_(d) slots isY _(s1) =H _(s) X _(s) +Z _(s1),  (134)where H_(s)∈

^(N) ^(s) ^(×M) is the static receiver channel which is known at thereceiver, and hence, X_(s) can be decoded. As a result, the achievabledegrees of freedom pair is

$\begin{matrix}{( {{N_{d}( {1 - \frac{N_{d}}{T_{d}}} )},\frac{N_{s}N_{d}}{T_{d}}} ),} & (135)\end{matrix}$which is strictly greater than TDMA. Thus, the product superpositionachieves nonzero degrees of freedom for the static receiver “for free”in the sense that the dynamic receiver achieves the single-receiverdegrees of freedom.

B. Achievability

Theorem 2:

Consider a K-receiver broadcast channel with heterogeneous coherencetimes and without CSIT or CSIR. The coherence times can be perfectlyaligned and integer multiples of each other, i.e.,

$\frac{T_{k}}{T_{k - 1}} \in {{\mathbb{Z}}.}$Define

⊆[1:K] to be a set of J receivers ordered ascendingly according to thecoherence time length. For j∈

, the set of degrees of freedom tuples

₁(

) can be achieved:

$\begin{matrix}{d_{j} = \{ {\begin{matrix}{{N_{j}^{*}( {1 - \frac{N_{j}^{*}}{T_{j}} - \frac{{\min\{ {M,N_{\max},T_{j}} \}} - N_{j}^{*}}{T_{j + 1}}} )},} & {j = j_{\min}} \\{{N_{j_{\min}}^{*}\min\{ {M,N_{j},T_{j_{\min}}} \}( {\frac{1}{T_{j - 1}} - \frac{1}{T_{j}}} )},} & {j \neq j_{\min}}\end{matrix}.} } & (136)\end{matrix}$Furthermore, the set of degrees of freedom tuples

₂(

) can be achieved:

$\begin{matrix}{d_{j} = \{ {\begin{matrix}{{N_{j}^{*}( {1 - \frac{N_{j}^{*}}{T_{j}}} )},} & {j = j_{\min}} \\{{N_{j_{\min}}^{*}\min\{ {N_{j},N_{j_{\min}}^{*}} \}( {\frac{1}{T_{j - 1}} - \frac{1}{T_{j}}} )},} & {j \neq j_{\min}}\end{matrix},{{{where}N_{j}^{*}} = {\min\{ {M,N_{j},{\frac{T_{j}}{2}}} \}}},{N_{\max} = {\max_{j \in {\mathbb{J}}}\{ N_{j} \}}}} } & (137)\end{matrix}$and j_(min) is the receiver with the shortest coherence time in

. The achievable degrees of freedom region is the convex hull of thedegrees of freedom tuples,

₁(

) and

(

), over all the possible sets

⊆[1:K], i.e.,

={(d ₁ , . . . ,d _(K))∈Co(

₁(

),

₂(

))),∀

⊆[1:K]}.  (138)The proof of Theorem 2 is provided below.

Remark 1:

The receivers of the set

are ordered ascendingly according to the coherence time length. J_(min)is the first receiver of

.

Remark 2:

The two achievable set of degrees of freedom tuples,

₁ (

) and

₂(

), are achieved by product superposition transmission scheme. Thedegrees of freedom gains are different in the two sets due to thedifference in the number of estimated antennas. In general, none of thetwo sets includes the other. In the first set,

₁(

), all the receivers estimate the channel of the maximum number ofantennas required for transmission, i.e., receiver j can estimate thechannel between N_(j)* antennas. In the second set,

₂(

), the receivers are limited to estimate the channel between N_(i1)antennas.

Remark 3:

When the receivers have the same coherence times, the productsuperposition scheme is not able to achieve degrees of freedom gains. Inthis case, there is no coherence diversity between the two receivers,hence, the degrees of freedom region is tight against TDMA.

C. Outer Bound

Theorem 3:

Consider a K-receiver broadcast channel under heterogeneous coherencetimes without CSIT or CSIR, meaning that the channel realization is notknown, but the channel distribution is globally known. The coherencetimes are perfectly aligned and integer multiples of each others, i.e.,

$\frac{T_{k}}{T_{k - 1}} \in {{\mathbb{Z}}.}$Define

⊆[1:K] to be a set of J receivers ordered ascendingly according to thecoherence time length, if a set of degrees of freedom tuples (d₁, . . ., d_(K)) is achievable, then it must satisfy the inequalities

$\begin{matrix}{{{\sum\limits_{j \in {\mathbb{J}}}\;\frac{d_{j}}{N_{j}^{*}( {1 - \frac{N_{j}^{*}}{T_{j_{\max}}}} )}} \leq 1},\mspace{14mu}{\forall{{\mathbb{J}} \subseteq \lbrack {1\text{:}K} \rbrack}},{{{where}\mspace{14mu} N_{j}^{*}} = {\min\{ {M,{N_{j}{\frac{T_{j}}{2}}}} \}}},} & (139)\end{matrix}$and j_(min) is the receiver with the longest coherence time in

.

Remark 4:

The receivers of the set

are ordered ascendingly according to the coherence time length, i.e.,

$\frac{T_{k}}{T_{k - 1}} \in {{\mathbb{Z}}.}$

_(max) is the last receiver of the set, and T_(j) _(max) is the longestcoherence time in the set

.

Proof:

To prove the Theorem, consider that for any

⊆[1:K], the degrees of freedom are bounded by the inequality of Eq.(139). First, show that for the set of receivers

, increasing the coherence time of the receivers to be equal to thelongest coherence time, i.e. T_(j)=T_(j) _(max) , ∀j∈

cannot reduce the degrees of freedom. The degrees of freedom region ofthe enhanced channel includes the original degrees of freedom region.Now, for

⊆[1:K], it can be shown that the degrees of freedom are non-decreasingwith the receivers coherence times.

Lemma 2:

For a K-receiver broadcast channel with heterogeneous coherence timesand without CSIT or CSIR, define

(

) to be the degrees of freedom region of a set of receivers

⊆[1:K] where the receivers are ordered ascendingly according to thecoherence time length. Define

(

) to be the degrees of freedom region of the same set of receivers

⊆[1:K] where the receivers have the coherence time of the longestreceiver, i.e., T_(j)=T_(j) _(max) , ∀j∈

. Thus,

$\begin{matrix}{{\mathcal{D}({\mathbb{J}})} \subseteq {\overset{\_}{\mathcal{D}}({\mathbb{J}})}} & (140)\end{matrix}$The proof of Lemma 2 is provided below. Using Lemma 2, the degrees offreedom region for every set of receivers

⊆[1:K] is included in the degrees of freedom region of an enhancedchannel with identical coherence times of length T_(j) _(max) slots.Furthermore, Theorem 1 shows that the degrees of freedom region of theenhanced channel is tight against TDMA inner bound. Thus, the region inEq. (139) is obtained, and hence, the proof of Theorem 3 is completed.

D. Optimality

For four cases, the achievable degrees of freedom region in SectionIII-B and the outer degrees of freedom region obtained in Section III-Care tight. In the four cases, the coherence time is at least twice thenumber of transmit and receive antennas, i.e., T_(j)≥2 max{M, N_(j)}.

Case 1—the Transmitter has Fewer Antennas:

When M≤min_(j){N_(j)}, the outer degrees of freedom region given by Eq.(139) is

$\begin{matrix}{{{\sum\limits_{j \in {\mathbb{J}}}\; d_{j}} \leq {M( {1 - \frac{M}{T_{j_{\max}}}} )}},\mspace{14mu}{\forall{{\mathbb{J}} \subseteq {\lbrack {1\text{:}K} \rbrack.}}}} & (141)\end{matrix}$The achievable degrees of freedom tuples in Eq. (137) are

$\begin{matrix}{\mspace{79mu}{d_{j} = \{ {\begin{matrix}{{M( {1 - \frac{M}{T_{j}}} )},} & {j = j_{\min}} \\{{M^{2}( {\frac{1}{T_{j - 1}} - \frac{1}{T_{j}}} )},} & {j \neq j_{\min}}\end{matrix},{j \in {{\mathbb{J}}.}}} }} & (142) \\{\mspace{79mu}{{Hence},{{\sum\limits_{j \in {\mathbb{J}}}\; d_{j}} = {{{M( {1 - \frac{M}{T_{j_{\min}}}} )} + {\sum\limits_{{j \in {\mathbb{J}}},{j \neq j_{\min}}}\;{M^{2}( {\frac{1}{T_{j - 1}} - \frac{1}{T_{j}}} )}}}\overset{(a)}{=}{{{M( {1 - \frac{M}{T_{j_{\min}}}} )} + {M^{2}( {\frac{1}{T_{j_{\min}}} - \frac{1}{T_{j_{\max}}}} )}} = {M( {1 - \frac{M}{T_{j_{\max}}}} )}}}},}} & (143)\end{matrix}$where (a) follows from the telescoping sum. Thus, the achievable degreesof freedom tuples are at the boundaries of the outer degrees of freedomregion, consequently, the convex hull of the achievable degrees offreedom tuples is tight against the outer degrees of freedom region.

Case 2—the Receivers have Equal Number of Antennas:

When N_(k)=N, ∀k, the outer degrees of freedom region given in Eq. (139)is

$\begin{matrix}{{{\sum\limits_{j \in {\mathbb{J}}}\; d_{j}} \leq {N^{*}( {1 - \frac{N^{*}}{T_{j_{\max}}}} )}},\mspace{14mu}{{\mathbb{J}} \subseteq {\lbrack {1\text{:}K} \rbrack.}}} & (144)\end{matrix}$The achievable degrees of freedom tuples in Eq. (137) are

$\begin{matrix}{\mspace{79mu}{d_{j} = \{ {\begin{matrix}{{N^{*}( {1 - \frac{N^{*}}{T_{j}}} )},} & {j = j_{\min}} \\{{N^{*2}( {\frac{1}{T_{j - 1}} - \frac{1}{T_{j}}} )},} & {j \neq j_{\min}}\end{matrix},{j \in {{\mathbb{J}}.}}} }} & (145) \\{\mspace{79mu}{{Hence},{{\sum\limits_{j \in {\mathbb{J}}}\; d_{j}} = {{{N^{*}( {1 - \frac{N^{*}}{T_{j_{\min}}}} )} + {\sum\limits_{{j \in {\mathbb{J}}},{j \neq j_{\min}}}\;{N^{*2}( {\frac{1}{T_{j - 1}} - \frac{1}{T_{j}}} )}}}\overset{(a)}{=}{{{N^{*}( {1 - \frac{N^{*}}{T_{j_{\min}}}} )} + {N^{*2}( {\frac{1}{T_{j_{\min}}} - \frac{1}{T_{j_{\max}}}} )}} = {{N^{*}( {1 - \frac{N^{*}}{T_{j_{\max}}}} )}.}}}}}} & (146)\end{matrix}$The achievable degrees of freedom tuples are at the boundaries of theouter degrees of freedom region, thus the outer degrees of freedomregion is tight.

Case 3—the Coherence Times of the Receivers are Very Large Compared tothe Coherence Time of One Receiver:

When T_(j)>>T₁, where j=2, . . . , K, the outer region given in Eq.(139) is

$\begin{matrix}{{d_{1} \leq {N_{1}^{*}( {1 - \frac{N_{1}^{*}}{T_{1}}} )}},{{\sum\limits_{j \in {\mathbb{J}}}\;\frac{d_{j}}{N_{j}^{*}}} \leq 1},\mspace{14mu}{{\mathbb{J}} \subseteq {\lbrack {1\text{:}K} \rbrack.}}} & (147)\end{matrix}$The achievable degrees of freedom tuples in Eq. (136),

₁(

), are

$\begin{matrix}{d_{j} = \{ {\begin{matrix}{{N^{*}( {1 - \frac{N^{*}}{T_{j}}} )},} & {j = j_{\min}} \\{\frac{N_{j_{\min}}^{*}N_{j}^{*}}{T_{j - 1}},} & {j \neq j_{\min}}\end{matrix},\mspace{14mu}{j \in {{\mathbb{J}}.}}} } & (148) \\{{Therefore},{{{\sum\limits_{j \in {\mathbb{J}}}\;\frac{d_{j}}{N_{j}^{*}}} \approx {1 - \frac{N_{j_{\min}}^{*}}{T_{j_{\min}}} + \frac{N_{j_{\min}}^{*}}{T_{j_{\min}}}}} = 1},} & (149)\end{matrix}$and the achievable degrees of freedom region is tight.

Case 4—The Receivers have Identical Coherence Time:

In the case of identical coherence times, it was shown in Section IIthat the degrees of freedom region is tight against TDMA. WhenT_(k)=T,∀k, the outer region given in Eq. (139) is

$\begin{matrix}{{{\sum\limits_{j \in {\mathbb{J}}}\;\frac{d_{j}}{N_{j}^{*}( {1 - \frac{N_{j}^{*}}{T}} )}} \leq 1},\mspace{14mu}{\forall{{\mathbb{J}} \subseteq \lbrack {1\text{:}K} \rbrack}},} & (150)\end{matrix}$which is the same as the TDMA degrees of freedom region. In this case,the achievable degrees of freedom tuples in Eq. (137),

₂(

), are reduced to TDMA.

E. Numerical Examples

Consider a single-antenna two-receiver broadcast channel, i.e. M=N₁=N₂=1with coherence times T₁=2 and T₂=4 slots. Thus, in this case, there arefour possibilities of

: { }, {1}, {2}, {1,2}. According to Theorem 3, the outer degrees offreedom region is given byd ₁≤½,d ₁ +d ₂≤¾.The achievable degrees of freedom tuples

$\begin{matrix}{{{{\mathbb{D}}_{1}({\mathbb{J}})} = {{{\mathbb{D}}_{2}({\mathbb{J}})}\text{:}\mspace{14mu}( {0,0} )}},( {\frac{1}{2},0} ),{( {0,\frac{3}{4}} ){( {\frac{1}{2},\frac{1}{4}} ).}}} & (151)\end{matrix}$FIG. 14 shows an example of the degrees of freedom region of atwo-receiver broadcast channel with heterogeneous coherence times whereM=N₁=N₂=1, T₁=2, T₂=4. As shown in FIG. 14, the outer and the achievableregions coincide on each other.

For a two-receiver broadcast channel with M=2, N₁=1, N₂=3, and T₁=4,T₂=24, the outer degrees of freedom is given by

${d_{1} \leq \frac{18}{24}},{{\frac{d_{1}}{23/24} + \frac{d_{2}}{44/24}} \leq 1.}$${Furthermore},{{{\mathbb{D}}_{1}({\mathbb{J}})}\text{:}\mspace{14mu}( {0,0} )},( {\frac{18}{24},0} ),( {0,\frac{44}{24}} ),( {\frac{17}{24},\frac{10}{24}} ),{{{\mathbb{D}}_{2}({\mathbb{J}})}\text{:}\mspace{14mu}( {0,0} )},( {\frac{18}{24},0} ),( {0,\frac{44}{24}} ),{( {\frac{18}{24},\frac{5}{24}} ).}$FIG. 15 shows an example of the degrees of freedom region of atwo-receiver broadcast channel with heterogeneous coherence times whereM=2, N₁=1, N₂=3, T₁=4, T₂=24. FIG. 15 illustrates the gap between theouter and the achievable bounds.

Furthermore, for a two-receiver broadcast channel with M=2, N₁=1, N₂=3and T₁=4 and T₂=40, the outer degrees of freedom region is given by

${d_{1} \leq \frac{30}{40}},{{\frac{d_{1}}{39/40} + \frac{d_{2}}{76/40}} \leq 1.}$For the achievable region in Theorem 2,

${{{\mathbb{D}}_{1}({\mathbb{J}})}\text{:}\mspace{14mu}( {0,0} )},( {\frac{30}{40},0} ),( {0,\frac{76}{40}} ),( {\frac{30}{40},\frac{9}{40}} )$${{{\mathbb{D}}_{2}({\mathbb{J}})}\text{:}\mspace{14mu}( {\frac{12}{16},0} )},( {0,\frac{28}{16}} ),{( {\frac{29}{40},\frac{18}{40}} ).}$FIG. 16 shows an example of the degrees of freedom region of atwo-receiver broadcast channel with heterogeneous coherence times whereM=2, N₁=1, N₂=3, T₁=4, T₂=40. FIG. 16 shows the gap between theachievable and the outer regions which is decreased with the increase ofthe ratio between the coherence times,

$\frac{T_{2}}{T_{1}}.$

Now consider a three-receiver broadcast channel with M=4, N_(z)=N₂=N₃=2and T₁=6, T₂=18, T₂=54. When the receivers have equal number ofantennas, as discussed in Section III-D, the achievable degrees offreedom and outer regions are tight. The outer degrees of freedom regionis

${d_{1} \leq \frac{5}{6}},{{\frac{d_{1}}{17/18} + \frac{d_{2}}{32/18}} \leq 1},{{\frac{d_{1}}{53/54} + \frac{d_{2}}{104/54} + \frac{d_{3}}{153/54}} \leq 1.}$For the achievable degrees of freedom region, there are 8 possibilitiesfor

:

-   -   { }, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}.        Hence,

${{{\mathbb{D}}_{1}({\mathbb{J}})}\text{:}\mspace{14mu}( {0,0,0} )},( {\frac{5}{6},0,0} ),( {0,\frac{32}{18},0} ),( {0,0,\frac{153}{154}} ),( {\frac{14}{18},\frac{4}{18},0} ),( {\frac{43}{54},0,\frac{24}{54}} ),( {0,\frac{94}{54},\frac{12}{54}} ),{{( {\frac{13}{18},\frac{4}{18},\frac{6}{54}} ).{{\mathbb{D}}_{2}({\mathbb{J}})}}\text{:}\mspace{14mu}( {0,0,0} )},( {\frac{5}{6},0,0} ),( {0,\frac{32}{18},0} ),( {0,0,\frac{153}{154}} ),( {\frac{5}{6},\frac{2}{18},0} ),( {\frac{5}{6},0,\frac{8}{54}} ),( {0,\frac{32}{18},\frac{8}{54}} ),{( {\frac{5}{6},\frac{2}{18},\frac{2}{54}} ).}$FIG. 17 shows an example of the degrees of freedom region of athree-receiver broadcast channel with heterogeneous coherence timeswhere M=4, N₁=N₂=N₃=2, T₁=8, T₂=24, T₂=72. FIG. 17 shows the achievabledegrees of freedom region (shaded area), the TDMA achievable region(dashed lines), and furthermore, the tight outer degrees of freedomregion (solid lines).

IV. Proof for Theorem 2

Achievable rates under coherence diversity for a general K-receiverbroadcast channel are attained by finding the best opportunities tore-use certain slots. Because the number of such opportunities blows upwith K, the central idea behind finding such opportunities are noteasily visible in the general case of K receivers, where the achievablerates are eventually described via an inductive process. To highlightthe ideas in the achievable rate methodology, these ideas are developedthrough an example of 3 receivers, which is the smallest number ofreceivers where the full richness of these interactions manifestthemselves. The K-receiver result is described in its full generality.

A. Achievability for Three Receivers

In the case of three receivers, there are 8 possible receivers sets J:one empty set { } achieving the trivial degrees of freedom tuple(0,0,0), three single receiver sets: {1}, {2}, {3}, three two-receiversets: {1,2}, {1,3}, {2,3}, and one three-receiver set {1,2,3}. In thesequel, the achievability of

₁(

) is first shown, followed by the achievability of

₂(

).

₁(

) Achievability:

For the three single-receiver sets, it is possible to achieve the threedegrees of freedom tuples

$\begin{matrix}{( {{N_{1}^{*}( {1 - \frac{N_{1}^{*}}{T_{1}}} )},0,0} ),( {0,{N_{2}^{*}( {1 - \frac{N_{2}^{*}}{T_{2}}} )},0} ),( {0,0,{N_{3}^{*}( {1 - \frac{N_{3}^{*}}{T_{3}}} )}} ),} & (152)\end{matrix}$by serving only one receiver while the other receivers remain unserved.In particular, for receiver k=1,2,3, every T_(k) slots, a trainingsequence is sent during N_(k)* slots and then data for receiver k issent during the remaining

$T_{k} - {N_{k}^{*}\mspace{14mu}{{slots}.\mspace{14mu}{N_{k}^{*}( {1 - \frac{N_{k}^{*}}{T_{k}}} )}}}$is achieved for receiver k whereas the other receivers achieve zerodegrees of freedom.

For the three two-receiver sets, two receivers are being served whilethe third receiver remains unserved. Using product superposition for tworeceivers, the degrees of freedom tuples are

$\begin{matrix}{( {{{N_{1}^{*}( {1 - \frac{N_{1}^{*}}{T_{1}}} )} - \frac{N_{1}^{*}( {{\min\{ {M,{\max\{ {N_{1},N_{2}} \}},T_{1}} \}} - N_{1}^{*}} )}{T_{2}}},{N_{1}^{*}\min\{ {M,N_{2},T_{1}} \}( {\frac{1}{T_{1}} - \frac{1}{T_{2}}} )},0} ),} & (153) \\{( {{{N_{1}^{*}( {1 - \frac{N_{1}^{*}}{T_{1}}} )} - \frac{N_{1}^{*}( {{\min\{ {M,{\max\{ {N_{1},N_{3}} \}},T_{1}} \}} - N_{1}^{*}} )}{T_{3}}},0,{N_{1}^{*}\min\{ {M,N_{3},T_{1}} \}( {\frac{1}{T_{1}} - \frac{1}{T_{3}}} )}} ),} & (154) \\{( {0,{{N_{2}^{*}( {1 - \frac{N_{2}^{*}}{T_{2}}} )} - \frac{N_{2}^{*}( {{\min\{ {M,{\max\{ {N_{2},N_{3}} \}},T_{2}} \}} - N_{2}^{*}} )}{T_{3}}},{N_{2}^{*}\min\{ {M,N_{3},T_{2}} \}( {\frac{1}{T_{2}} - \frac{1}{T_{3}}} )}} ).} & (155)\end{matrix}$To achieve Eq. (153), product superposition transmission is as follows.

-   -   Every T₂-length interval is divided into T₁-length subintervals.    -   During the first subinterval, a training sequence is sent during        min{M, max{N₁, N₂}, T₁} slots for receiver 1 and receiver 2        channel estimation. After that, data for receiver 1 is sent        during the following T₁−min{M, max{N₁, N₂}, T₁} slots. Receiver        1 achieves N₁*(T₁−min{M, max{N₁, N₂}, T₁}) degrees of freedom.    -   During the remaining subintervals, the transmitter sends, every        T₁ slots,

$\begin{matrix}{{X_{i}^{(12)} = \lbrack {V_{i},{V_{i}U_{i}}} \rbrack},\mspace{14mu}{i = 1},\ldots\mspace{14mu},{\frac{T_{2}}{T_{1}} - 1},} & (156)\end{matrix}$

-   -   where U_(i)∈        ^(N) ¹ ^(*) ^(×(T) ¹ ^(−N) ¹ ^(*) ), V_(i) ∈        ^(M×N) ₁* are data matrices for receiver 1, and receiver 2,        respectively. Thus, receiver 1 estimates its equivalent channel        H _(1,i)=H_(1,i)V_(i), and decodes U_(i) achieving

$( {\frac{T_{2}}{T_{1}} - 1} ){N_{1}^{*}( {T_{1} - N_{1}^{*}} )}$degrees of freedom. Furthermore, the channel of receiver 2 remainsconstant and known, hence, V_(i) can be decoded coherently at receiver 2achieving

$( {\frac{T_{2}}{T_{1}} - 1} )N_{1}^{*}\min\{ {M,N_{2},T_{1}} \}$degrees of freedom since when N₂≥T₁, receiver 2 estimates only T₁antennas during the first subinterval.Thus, by the above product superposition scheme, for every T₂ slots,receiver 1 achieves

${\frac{T_{2}}{T_{1}}{N_{1}^{*}( {T_{1} - N_{1}^{*}} )}} - {N_{1}^{*}( {{\min\{ {M,{\max\{ {N_{1},N_{2}} \}},T_{1}} \}} - N_{1}^{*}} )}$degrees of freedom, and furthermore, receiver 2 achieves

$( {\frac{T_{2}}{T_{1}} - 1} )N_{1}^{*}\min\{ {M,N_{2},T_{1}} \}$degrees of freedom obtaining Eq. (153).

For achieving (154), product superposition transmission is as follows.

-   -   Every T₃-length interval is divided into T₁-length subintervals.    -   During the first subinterval, a training sequence is sent during        min{M, max{N₁, N₃}, T₁} slots for receiver 1 and receiver 3        channel estimation. After that, data for receiver 1 is sent        during the following T₁−min{M, max{N₁, N₃}, T₁} slots. Receiver        1 achieves N₁*(T₁−min{M, max{N₁, N₃}, T₁}) degrees of freedom.    -   During the remaining subintervals, every T₁ slots, the        transmitted signal is similar to Eq. (156), yet, after replacing        V_(i) with W_(i) which contains data for receiver 3. Receiver 1,        and receiver 3 achieve

$( {\frac{T_{3}}{T_{1}} - 1} ){N_{1}^{*}( {T_{1} - N_{1}^{*}} )}$and

$( {\frac{T_{3}}{T_{1}} - 1} )N_{1}^{*}\min\{ {M,N_{3},T_{1}} \}$degrees of freedom, respectively.Thus, for every T₃ slots, receiver 1 achieves

${\frac{T_{3}}{T_{1}}{N_{1}^{*}( {T_{1} - N_{1}^{*}} )}} - {N_{1}^{*}( {{\min\{ {M,{\max\{ {N_{1},N_{3}} \}},T_{1}} \}} - N_{1}^{*}} )}$degrees of freedom, and furthermore, receiver 3 achieves

$( {\frac{T_{3}}{T_{1}} - 1} )N_{1}^{*}\min\{ {M,N_{3},T_{1}} \}$degrees of freedom obtaining Eq. (154).

Similar to the degrees of freedom tuples (38) and (39), we can achieve(40) by the same transmission strategy, yet, with respect to T₂ and T₃.Thus, every T₃ slots, receiver 2 achieves

${\frac{T_{3}}{T_{2}}{N_{2}^{*}( {T_{2} - N_{2}^{*}} )}} - {N_{2}^{*}( {{\min\{ {M,{\max\{ {N_{2},N_{3}} \}},T_{2}} \}} - N_{2}^{*}} )}$degrees of freedom, and furthermore, receiver 3 achieves

$( {\frac{T_{3}}{T_{2}} - 1} )N_{1}^{*}\min\{ {M,N_{3},T_{2}} \}$degrees of freedom.

Now the remaining degrees of freedom tuple is the one with thethree-receiver set {1,2,3}. In this case, the achievable degrees offreedom tuples are

$( {{{N_{1}^{*}( {1 - \frac{N_{1}^{*}}{T_{1}}} )} - \frac{N_{1}^{*}( {{\min\{ {M,{\max\{ {N_{1},N_{2},N_{3}} \}},T_{1}} \}} - N_{1}^{*}} )}{T_{2}}},{N_{1}^{*}\min\{ {M,N_{2},T_{1}} \}( {\frac{1}{T_{1}} - \frac{1}{T_{2}}} )},{N_{1}^{*}\min\{ {M,N_{3},T_{1}} \}( {\frac{1}{T_{2}} - \frac{1}{T_{3}}} )}} ),$which can be achieved by product superposition as follows.

-   -   Every T₃-length interval is divided into T₂-length subintervals.    -   During the first subinterval, the transmitted signal is the same        as that used to achieve Eq. (153). Thus, receiver 1 achieves

${\frac{T_{2}}{T_{1}}{N_{1}^{*}( {T_{1} - N_{1}^{*}} )}} - {N_{1}^{*}( {{\min\{ {M,{\max\{ {N_{1},N_{2}} \}},T_{1}} \}} - N_{1}^{*}} )}$degrees of freedom, receiver 2 achieves

$( {\frac{T_{2}}{T_{1}} - 1} )N_{1}^{*}\min\{ {N_{2}^{*},T_{1}} \}$degrees of freedom, and furthermore, receiver 3 estimates its channel.

-   -   During the remaining

$( {\frac{T_{3}}{T_{2}} - 1} )$subintervals, the transmitter sends, every T₂-length subinterval, thesame signal that achieves Eq. (153) after multiplying it from the leftby W_(i) which contains data for receiver 3. Therefore, during the firstT₁ of every T₂-length subinterval, the transmitted signal isX ⁽¹²³⁾=[W _(i) ,W _(i) U _(i)].  (157)

-   -   After that during

$( {\frac{T_{2}}{T_{1}} - 1} )T_{1}$slots, the transmitted signal is{tilde over (X)} ⁽¹²³⁾=[W _(i) V _(i) ,W _(i) V _(i) U _(i)].  (158)

-   -   Thus, receiver 1 can estimate the equivalent channel H        _(1,i)=H_(1,i)W_(i)V_(i), and decodes U_(i), receiver 2 can        estimate H _(2,i)=H_(2,i)W_(i), and can decode V_(i) and        receiver 3 can decode W_(i). Thus, receiver 1, receiver 2, and        receiver 3 achieve

${( {\frac{T_{3}}{T_{2}} - 1} )\frac{T_{2}}{T_{1}}{N_{1}^{*}( {T_{1} - N_{1}^{*}} )}},{( {\frac{T_{3}}{T_{2}} - 1} )( {\frac{T_{2}}{T_{1}} - 1} )N_{1}^{*}\min\{ {M,N_{2},T_{1}} \}\mspace{14mu}{and}\mspace{14mu}( {\frac{T_{3}}{T_{2}} - 1} )N_{1}^{*}\min\{ {M,N_{3},T_{1}} \}}$degrees of freedom, respectively.Therefore, by using the above product superposition transmission scheme,the above degrees of freedom tuple can be achieved.

₂(

) Achievability:

Similar to

₁(

), it is possible to achieve the degrees of freedom tuples of Eq. (152)that correspond to the three single-receiver sets by serving only onereceiver while the other receivers remain unserved.

For the three two-receiver sets, only two receivers are being servedwhile the third receiver remains unserved. Using product superpositionfor two receivers, the degrees of freedom tuples

$\begin{matrix}{( {{N_{1}^{*}( {1 - \frac{N_{1}^{*}}{T_{1}}} )},{N_{1}^{*}\min\{ {N_{1}^{*},N_{2}} \}( {\frac{1}{T_{1}} - \frac{1}{T_{2}}} )},0} ),} & (159) \\{( {{N_{1}^{*}( {1 - \frac{N_{1}^{*}}{T_{1}}} )},0,{N_{1}^{*}\min\{ {N_{1}^{*},N_{3}} \}( {\frac{1}{T_{1}} - \frac{1}{T_{3}}} )}} ),} & (160) \\{( {0,{N_{2}^{*}( {1 - \frac{N_{2}^{*}}{T_{2}}} )},{N_{2}^{*}\min\{ {N_{2}^{*},N_{3}} \}( {\frac{1}{T_{2}} - \frac{1}{T_{3}}} )}} ),} & (161)\end{matrix}$are achieved as follows. To achieve Eq. (159), the product superpositiontransmission strategy is as follows.

-   -   Every T₂-length interval is divided into T₁-length subintervals.    -   During the first T₁-length subinterval, a training sequence is        sent during N₁* slots and data for receiver 1 is sent during the        following T₁−N₁* slots. Receiver 1 achieves N₁*(T₁−N₁*) degrees        of freedom, and receiver 2 estimates its channel between        min{N₁*, N₂} transmit antennas.    -   During the remaining subintervals, every T₁ slots, the        transmitter sends

$\begin{matrix}{{X_{i}^{(12)} = \lbrack {V_{i},{V_{i}U_{i}}} \rbrack},\mspace{14mu}{i = 1},\ldots\mspace{14mu},{\frac{T_{2}}{T_{1}} - 1},} & (162)\end{matrix}$

-   -   Thus, receiver 1, and receiver 2 achieve

${( {\frac{T_{2}}{T_{1}} - 1} ){N_{1}^{*}( {T_{1} - N_{1}^{*}} )}},{{and}\mspace{14mu}( {\frac{T_{2}}{T_{1}} - 1} )N_{1}^{*}\min\{ {N_{1}^{*},N_{2}} \}}$degrees of freedom, respectively.Thus, by the above product superposition scheme, for every T₂ slots,receiver 1, and receiver 2 achieve

${\frac{T_{2}}{T_{1}}{N_{1}^{*}( {T_{1} - N_{1}^{*}} )}},{{and}\mspace{14mu}( {\frac{T_{2}}{T_{1}} - 1} )N_{1}^{*}\min\{ {N_{1}^{*},N_{2}} \}}$degrees of freedom, respectively obtaining Eq. (159).

For achieving Eq. (160) by product superposition, the same transmissionscheme of achieving Eq. (159) can be used with respect to receiver 1 andreceiver 3, i.e. replacing T₂, min{N₁*, N₂} with T₃, min{N₁*, N₃},respectively. Thus, receiver 1, and receiver 3 achieve

${\frac{T_{3}}{T_{1}}{N_{1}^{*}( {T_{1} - N_{1}^{*}} )}},{{and}\mspace{14mu}( {\frac{T_{3}}{T_{1}} - 1} )N_{1}^{*}\min\{ {N_{1}^{*},N_{3}} \}}$degrees of freedom, respectively, for every T₃ slots. Similarly, Eq.(161) can be achieved by the same transmission strategy, yet, withrespect to T₂ and T₃.

For the three-receiver set, the achievable degrees of freedom tuples are

$( {{N_{1}^{*}( {1 - \frac{N_{1}^{*}}{T_{1}}} )},{N_{1}^{*}\min\{ {N_{1}^{*},N_{2}} \}( {\frac{1}{T_{1}} - \frac{1}{T_{2}}} )},{N_{1}^{*}\min\{ {N_{1}^{*},N_{3}} \}( {\frac{1}{T_{2}} - \frac{1}{T_{3}}} )}} ),$which can be achieved by product superposition for the three receiversas follows.

-   -   Every T₃-length interval is divided into T₂-length subintervals.    -   During the first subinterval, the transmitted signal is the same        as that used to achieve Eq. (159). Thus, receiver 1, and        receiver 2 achieves

${\frac{T_{2}}{T_{1}}{N_{1}^{*}( {T_{1} - N_{1}^{*}} )}},$and

$( {\frac{T_{2}}{T_{1}} - 1} )N_{1}^{*}\min\{ {N_{1}^{*},N_{2}} \}$degrees of freedom, and furthermore, receiver 3 estimates its channelbetween min{N₁*, N₃} transmit antennas.

-   -   During the remaining

$( {\frac{T_{3}}{T_{2}} - 1} )$subintervals, the transmitter sends, every T₂-length subinterval, thesame signal that achieves Eq. (159) after multiplying it from the leftby W_(i) which contains data for receiver 3. Therefore, during the firstT₁ of every T₂-length subinterval, the transmitted signal isX ⁽¹²³⁾=[W _(i) ,W _(i) U _(i)].  (163)

-   -   After that during

$( {\frac{T_{2}}{T_{1}} - 1} )$T₁ slots, the transmitted signal is{tilde over (X)} ⁽¹²³⁾=[W _(i) V _(i) ,W _(i) V _(i) U _(i)].  (164)

-   -   Thus, receiver 1 can estimate the equivalent channel H        _(1,i)=H_(1,i)W_(i)V_(i), and decodes U_(i). Also, receiver 2        can estimate H _(2,i)=H_(2,i)W_(i), and decode V_(i) and        furthermore, receiver 3 can decode W_(i). Receiver 1, receiver        2, and receiver 3 achieve

${( {\frac{T_{3}}{T_{2}} - 1} )\frac{T_{2}}{T_{1}}{N_{1}^{*}( {T_{1} - N_{1}^{*}} )}},{( {\frac{T_{3}}{T_{2}} - 1} )( {\frac{T_{2}}{T_{1}} - 1} )N_{1}^{*}\min\{ {N_{1}^{*},N_{2}} \}\mspace{14mu}{{and}( {\frac{T_{3}}{T_{2}} - 1} )}N_{1}^{*}\min\{ {N_{1}^{*},N_{3}} \}}$degrees of freedom, respectively.Therefore, the above degrees of freedom tuples can be obtained.

B. Achievability for K Receivers

To obtain the achievability for the K-receiver case, it can be shownthat for every set of receivers

⊆[1:K], ordered ascendingly according to the coherence time length, thedegrees of freedom tuples

₁(

) and

₂(

) are achievable. An induction argument is used in the proof as follows.First, the achievability when

has only three receivers was shown in Section IV. In the sequel, therest of the proof is dedicated to show that for arbitrary set ofreceivers,

⊆[1:K] where the receivers are ordered ascendingly according to thecoherence time length, the product superposition achieves the degrees offreedom tuples

₁(

)/

₂(

), it is possible to achieve the degrees of freedom tuple

₁(

)/

₂(

), where

⊆[1:K] is the set constructed by adding one more receiver to the set

where the length of the added receiver coherence time is an integermultiple of j_(max). To complete the proof it can be shown that productsuperposition achieves the degrees of freedom tuples

₁(

)/

₂(

) for the set

. The following Lemma addresses this part of the proof.

Lemma 3:

For the broadcast channel considered in Section III, define X_(O)∈

^(T) ^(τ) ^(×T) ^(o) to be a pilot-based transmitted signal during T_(o)slots where a training matrix X_(τ)∈

^(T) ^(τ) ^(×T) ^(τ) is sent during T_(τ) slots and then the data issent during T_(O)−T_(τ) slots achieving the degrees of freedom tuple

^((o))=(d₁ ^((o)), d₂ ^((o)), . . . , d_(J) ^((o))) for J receivers. Itis possible to achieve

^((o)) for the J receivers in addition to

$( {\frac{T_{\epsilon}}{T_{O}} - 1} )\frac{T_{\tau}\min\{ {T_{\tau}N_{\epsilon}^{*}} \}}{T_{\epsilon}}$to a receiver ∈ with T_(∈)-length coherence time and N_(∈) receiveantennas, where

$\frac{T_{\epsilon}}{T_{O}} \in {{\mathbb{Z}}.}$

Proof:

This can be achieved by the following product superposition transmissionscheme.

-   -   Every T_(∈) slots is divided into T_(O)-length subintervals.    -   During the first T_(O) subinterval, the transmitted signal is        X_(O). Thus        ^((o)) degrees of freedom tuple is achieved for the J receivers        and no degrees of freedom for the receiver ∈, yet, it estimates        its channel between min{N_(∈)*, T_(τ)} transmission antennas.    -   During the remaining subintervals, product superposition is        used. Every T_(O) slots, the transmitter sends        {tilde over (X)} _(o) =PX _(o),  (165)    -   where P∈        ^(M×T) ^(τ) contains data for receiver ∈. X_(O) contains the        training matrix X_(τ), hence, receiver ∈ can decode P, using its        channel estimate. Furthermore, the J receivers estimate their        equivalent channels and decode their data during T_(O)−T_(τ)        slots. Thus, the J receivers achieve

$( {\frac{T_{\epsilon}}{T_{O}} - 1} )\mathcal{D}^{(o)}$degrees of freedom tuple and receiver ∈ achieves

$( {\frac{T_{\epsilon}}{T_{O}} - 1} )T_{\tau}\min\{ {N_{\epsilon}^{*},T_{\tau}} \}$degrees of freedom.Thus, in T_(∈) slots, J receivers achieve

$\frac{T_{\epsilon}}{T_{O}}\mathcal{D}^{(o)}$degrees of freedom, and furthermore, receiver ∈ achieves

$( {\frac{T_{\epsilon}}{T_{O}} - 1} )T_{\tau}\min\{ {N_{\epsilon}^{*},T_{\tau}} \}$degrees of freedom which completes the proof of Lemma 3. Using Lemma 3the second part of the proof is completed, and hence, the proof ofTheorem 2 is completed.

V. General Coherence Times

In this section, a K-receiver broadcast channel with general coherencetimes is examined. Achievable degrees of freedom region can be obtained,where the coherence times have arbitrary ratio or alignment. Coherencetimes, as is required in a block fading model in a time-sampled domain,continue to take positive integer values.

A. Unaligned Coherence Times

In this section, the assumption on the alignment of coherence intervalsis relaxed. Consider a broadcast channel with K receivers where thecoherence times are integer multiple of each other, i.e.

$\frac{T_{k}}{T_{k - 1}} \in {{\mathbb{Z}}.}$The coherence times have arbitrary alignment, meaning that there couldbe an offset between the transition times of the coherence intervals ofdifferent receivers. Recall that in the case of aligned coherenceintervals, product superposition provided the achievable degrees offreedom region in Eq. (138). The receiver with longer coherence timereuses some of the unneeded pilots and achieves gains in degrees offreedom without affecting the receivers with shorter coherence times.Under unaligned coherence times the same gains in degrees of freedom areavailable with product superposition. Using the transmitted signal givenin Section IV, the longer coherence times include the same number ofunneeded pilot sequences regardless of the alignment. These unneededpilot sequences can be reused by product superposition transmission,achieving degrees of freedom gain. For instance, if there are tworeceivers with M=2, N₁=N₂=1, T₁=4, T₂=8, with an offset of onetransmission symbol as shown in FIG. 18, which illustrates productsuperposition transmission for unaligned coherence times. It is possibleto achieve the degrees of freedom pair

$( {\frac{3}{4},\frac{1}{8}} )$via a transmission strategy over pairs of coherence intervals forreceiver 1, as follows.

-   -   In odd intervals for receiver 1, one pilot is transmitted during        which both receivers estimate their channels. In the 3 remaining        times in this interval, data is transmitted for receiver 1.    -   In the even intervals for receiver 1, during the first time slot        a product superposition is transmitted providing one degree of        freedom for receiver 2 (whose channel has not changed) while        allowing receiver 1 to renew the estimate of his channel. The        three remaining time slots provide 3 further degrees of freedom        for receiver 1.        Thus, in 8 time slots, receiver 1 achieved 6 degrees of freedom        and receiver 2 achieved 1. This is the same “corner point” that        was obtained earlier, noting that the nature of the algorithm is        not changed, only the position of the pilot transmission must be        carefully chosen while keeping in mind the transition points of        the block fading.

B. Unaligned Coherence Times with Perfect Symmetry (Staggered)

Now consider a special case of two-receiver unaligned coherence timeswhere the transition of each coherence interval is exactly in the middleof the other coherence interval. This special case is motivated by blindinterference alignment model in FIG. 19, and for easy reference we callthis configuration a staggered coherence time. FIG. 19 shows blindinterference alignment for staggered coherence times with CSIR.

Follow the example of blind interference alignment: a 2-receiverbroadcast channel with M=2, N₁=N₂=1. FIG. 20 illustrates blindinterference alignment with pilot transmission. Receiver 1 cancelsh_(1i) ^(H)v, and decodes u, whereas receiver 2 cancels h_(2i) ^(H)u,and decodes v. As shown in FIG. 20, the transitions of the longercoherence interval occur at the middle of the shorter coherenceinterval. Based on the discussion in Section V-A, product superpositioncan obtain degrees of freedom gain for the staggered scenario. Blindinterference alignment can achieve degrees of freedom pair

$( {\frac{2}{3},\frac{2}{3}} )$while ignoring the cost of CSIR. To allow comparison and synergy, aversion of blind interference alignment with channel estimation shown inFIG. 20 is analyzed. The gain of blind interference alignment comes fromthe staggering of the coherence time, whereas the source of productsuperposition gain is reusing the unneeded pilots with respect to thelonger coherence times. Therefore, it is possible to give a transmissionscheme that uses both blind interference alignment and productsuperposition over

$\frac{T_{2}}{T_{1}}$coherence intervals of receiver 1, as shown in FIG. 21.

-   -   During the first 2 intervals, two pilots are sent at the middle        of each interval for channel estimation. Furthermore, during the        first interval and half of the second interval,

$( {\frac{T_{1}}{2} - 1} )$blind interference alignment signaling is sent achieving

$( {{2( {\frac{T_{1}}{2} - 1} )},{2( {\frac{T_{1}}{2} - 1} )}} )$degrees of freedom pair. For the remaining half of the second interval,data for receiver 1 is sent achieving

$( {\frac{T_{1}}{2} - 1} )$further degrees of freedom.

-   -   During the remaining

$( {\frac{T_{2}}{T_{1}} - 1} )$intervals, product superposition transmission is sent achieving

$( {{( {\frac{T_{2}}{T_{1}} - 1} )( {T_{1} - 1} )},( {\frac{T_{2}}{T_{1}} - 1} )} )$degrees of freedom pair.

-   -   Receiver 2 estimates its channel during the first and the last        time slots of its coherence interval.        Thus, the above transmission scheme obtain the degrees of        freedom pair

$\begin{matrix}{( {{1 - \frac{1}{T_{1}} - \frac{1}{T_{2}} - \frac{T_{1}}{2\; T_{2}}},{\frac{T_{1}}{T_{2}} + \frac{1}{T_{1}} - \frac{4}{T_{2}}}} ).} & (166)\end{matrix}$Furthermore, product superposition can achieve the degrees of freedompair

$( {{1 - \frac{1}{T_{1}}},{\frac{1}{T_{2}} - \frac{1}{T_{1}}}} )$Hence, the achievable degrees of freedom is the convex hull of thedegrees of freedom pairs achieved by blind interference alignment,product superposition, and combining blind interference alignment withproduct superposition.

C. Arbitrary Coherence Times

Theorem 4:

Consider a K-receiver broadcast channel without CSIT or CSIR havingheterogeneous coherence times, where the coherence times are allowed totake any positive integer value. Product superposition can achieve thedegrees of freedom tuple defined in Eq. (137).

Remark 5:

Blind interference alignment signaling can be sent at the location ofthe staggering coherence times achieving degrees of freedom gain. Hence,similar to the case of staggered coherence times with integer ratio inSection V-B, product superposition can be combined with blindinterference alignment increasing the achievable degrees of freedomregion.

Proof:

For clarity of explanation, we start by giving the achievable scheme for3 receivers with

${N_{k} = {N \leq {\min\{ {M,{\frac{T_{1}}{2}}} \}}}},$∀k over (T₂T₃) coherence intervals of receiver 1.

-   -   For every interval, a pilot sequence of length N slots, and        receiver 1 data of length T₁−N slots are sent, achieving N(T₁−N)        degrees of freedom for receiver 1.    -   The number of pilot sequences of length N slots is T₂T₃. Having        coherence time T₂, receiver 2 needs only T₁T₃ pilot sequences        for channel estimation. Hence, produced superposition can be        sent during (T₂T₃−T₁T₃) pilot sequences to send data for        receiver 2 achieving NT₃(T₂−T₁) degrees of freedom.    -   Furthermore, receiver 3 needs only T₁T₂ pilot sequences for        channel estimation, and hence, data signal for receiver 3 can be        sent during (T₂T₃−T₁T₂) pilot sequences via product        superposition. Since, data for receiver 2 already is sent via        product superposition during (T₂T₃−T₁T₃) pilot sequences,        receiver 3 can only reuse (T₂T₃−T₁T₂)−(T₂T₃−T₁T₃)=(T₃−T₂)T₁        pilot sequences achieving NT(T₃−T₂) degrees of freedom.        Thus, it is possible to achieve the degrees of freedom tuple

$\begin{matrix}{( {{N( {1 - \frac{N}{T_{1}}} )},{N^{2}( {\frac{1}{T_{1}} - \frac{1}{T_{2}}} )},{N^{2}( {\frac{1}{T_{2}} - \frac{1}{T_{3}}} )}} ).} & (167)\end{matrix}$

Now, consider the proof for arbitrary number of receivers, and generalantenna setup. For a set of receiver

⊆[1:K] having J receiver where

${\frac{T_{j}}{T_{j - 1}} \in {\mathbb{Q}}},$j ∈

the degrees of freedom tuple of Eq. (137) can be obtained over Π_(i=2)^(J)T_(i) coherence intervals of receiver j_(min).

-   -   For every interval, a pilot sequence of length N_(j) _(min) *        slots, and data of length T_(j) _(min) −N_(j) _(min) * for        receiver j_(min) are sent, achieving N_(j) _(min) *(T_(j) _(min)        −N_(j) _(min) *) degrees of freedom for receiver j_(min).    -   The number of pilot sequences of length N_(j) _(min) * slots is        Π_(i=2) ^(J)T_(i). Receiver j=j_(min), with coherence time        T_(j), can estimate the channel of min{N_(j) _(min) *, N_(j)}        transmit antennas using Π_(i=1,i≠j) ^(J)T_(i) pilot sequences.        After excluding the pilots reused by receivers i={j_(min)+1, . .        . , j−1} to send data by product superposition transmission,        data for receiver j can be sent via product superposition during        (T_(j)−T_(j−1))Π_(i=1,i≠j,j−1) ^(J)T_(i) pilots obtaining the        degrees of freedom N_(j) _(min) *min{N_(j) _(min)        *,N_(j)}(T_(j)−T_(j−1))Π_(i=1,i≠j,j−1) ^(J)T_(i).        Thus, the proof of Theorem 4 is completed.

VI. Multiple Access Channel with Identical Coherence Times

Consider a K-transmitter MIMO multiple access channel without CSIT orCSIR, where transmitter k is equipped with M_(k) antennas, and thereceiver is equipped with N antennas. The received signal at thediscrete time n can be given by

$\begin{matrix}{{{y(n)} = {{\sum\limits_{k = 1}^{K}{{{\overset{\_}{H}}_{k}(n)}{x_{k}(n)}}} + {z(n)}}},} & (168)\end{matrix}$where x_(k)(n)∈

^(M) ^(k) ^(×1) is transmitter k transmitted signal, z(n)∈

^(N×1) is the i.i.d. Gaussian additive noise and H _(k)(n)∈

^(N×M) ^(k) is transmitter k Rayleigh block-fading channel matrix withcoherence time T_(k). Consider the case when T_(k)≥2N, ∀k.

Assume that all transmitters have identical coherence time, T. In thesequel, define a degrees of freedom achievable region based on apilot-based scheme in Section VI-A. Furthermore, an outer degrees offreedom region is given in Section VI-B based on the cooperative bound.Some numerical examples are given in Section VI-C where it is shown thatthe achievable and the outer degrees of freedom regions coincide at theregions of sum degrees of freedom.

A. Achievability

Theorem 5:

Consider a K-transmitter MIMO multiple access channel without CSIT orCSIR, meaning that the channel realization is not known, but the channeldistribution is globally known. If the transmitters have identicalcoherence times, namely T, then for every ordered set of transmitters,

={k₁, k₂, . . . , k_(j)}⊆[1:K], we can achieve the set of degrees offreedom tuple

(

):

$\begin{matrix}{{d_{j} = {M_{j}^{\prime}( {1 - \frac{M_{j}^{\prime}}{T}} )}},{j \in},} & (169)\end{matrix}$where M_(j)′=min{M_(j),[N−Σ_(m=1) ^(j−1)M_(k) _(m) ′]⁺}, and T≥2N. Theachievable degrees of freedom region is the convex hull of the degreesof freedom tuples,

(

), overall the

$\sum\limits_{i = 1}^{K}\frac{K!}{( {K - i} )!}$Possible ordered sets

⊆[1:K], i.e.,

={(d ₁ , . . . ,d _(K))∈Co(

(

)),∀

⊆[1:K]}.  (170)

Proof:

It can be shown that a simple pilot-based scheme can achieve the aboveachievable degrees of freedom region. Assume an ordered set oftransmitters

={k₁, k₂, . . . , k_(J)}⊆[1:K]. In order to achieve the degrees offreedom tuple in (169), the following transmission scheme can be used.

-   -   At the beginning of every T-length interval, a training sequence        of length        M_(j)′ is sent so that the receiver can estimate the channels        where a M_(j)′-length training sequence is sent from transmitter        j.    -   During the remaining T−        M_(j)′ period, simultaneously, M_(j)′×(T−        M_(j)′) data matrix is sent from transmitter j. Hence, the        receiver, using        M_(j)′ antennas, can invert the channel (via zero forcing) and        decode the transmitted signal during T−        M_(j)′; slots.        Therefore, every T period, transmitter j∈        can achieve M_(j)′(T−        M_(j)′) degrees of freedom, and hence Eq. (169) can be obtained.

B. Outer Bound

For the considered K-transmitter multiple access channel with identicalcoherence times, namely T, the cooperative bound can be given by

$\begin{matrix}{{{\sum\limits_{j \in {\mathbb{J}}}R_{j}} \leq {I( {{X({\mathbb{J}})}; Y \middle| {X( {\mathbb{J}}^{c} )} } )}},{\forall{{\mathbb{J}} \subseteq {\lbrack {1:K} \rbrack.}}}} & (171)\end{matrix}$An outer bound on the degrees of freedom region is,

$\begin{matrix}{{{\sum\limits_{j \in {\mathbb{J}}}d_{j}} \leq {\min\{ {N,{\sum\limits_{j \in {\mathbb{J}}}M_{j}}} \}( {1 - \frac{\min\{ {N,{\sum\limits_{j \in {\mathbb{J}}}M_{j}}} \}}{T}} )}},} & (172)\end{matrix}$

C. Numerical Examples

Consider a two-transmitter multiple access channel where thetransmitters are equipped with M₁=3, M₂=2 antennas and the receiver isequipped with N₂=4 antennas. The coherence time of the two transmittersis T=10 slots. Thus, the outer degrees of freedom region is given by

$\begin{matrix}{{d_{1} \leq \frac{21}{10}},} \\{{d_{2} \leq \frac{16}{10}},} \\{{d_{1} + d_{2}} \leq {\frac{24}{10}.}}\end{matrix}$The achievable degrees of freedom pairs in Theorem 5 can be obtained asfollows. For the case of two transmitters, there are 5 ordered sets oftransmitters

: { }, {1}, {2}, {1,2} and {2,1}. For { }, the trivial degrees offreedom pair (0,0) can be obtained. For the two sets {1}, {2}, thedegrees of freedom pairs

$( {\frac{21}{10},0} )$and

$( {0,\frac{16}{10}} ),$respectively, can be obtained. For the two sets {1,2} and {2,1}, thedegrees of freedom pairs

$( {\frac{18}{10},\frac{6}{10}} )$and

$( {\frac{12}{10},\frac{12}{10}} ),$respectively, can be obtained. FIG. 22 shows an example of degrees offreedom region of a two-transmitter multiple access channel withidentical coherence times where M₁=3, M₂=2, N=4, T=10. The convex hullof the achieved degrees of freedom pairs gives the achievable degrees offreedom region which is tight against the sum degrees of freedom asshown in FIG. 22.

Consider a two-transmitter multiple access channel where thetransmitters are equipped with M₁=4, M₂=2 antennas and the receiver isequipped with N=3 antennas. The coherence time of the two transmittersis T=10 slots. FIG. 23 shows an example of degrees of freedom region ofa two-transmitter multiple access channel with identical coherence timeswhere M₁=4, M₂=2, N=3, T=10. As shown in FIG. 23, the achievable degreesof freedom regions are tight against the sum degrees of freedom.

VII. Multiple Access Channel with Heterogeneous Coherence Times

Consider the multiple access channel defined in (53) where there is noCSIT or CSIR. Consider the case where the receivers coherence times areperfectly aligned and integer multiples of each other, i.e., ∀k,

$\frac{T_{k}}{T_{k - 1}} \in .$In the sequel, an achievable degrees of freedom region and an outerdegrees of freedom region are given in Section VII-A and Section VII-B,respectively. Also, some numerical examples are given in Section VII-Cto demonstrate the achievable and the outer degrees of freedom regions.

A. Achievability

Theorem 6:

Consider a K-transmitter MIMO multiple access channel without CSIT orCSIR, meaning that the channel realization is not known, but the channeldistribution is globally known. Furthermore, the transmitters coherencetimes are assumed to be perfectly aligned and integer multiples of eachother. Define

={i₁, . . . , i_(j)}⊆[1:K] to be a set of J transmitters where ∀j∈

,

${\frac{T_{j}}{T_{j - 1}} \in},$Define

={k₁, . . . , k_(j)} to be one of the J!possible ordered sets of

. If T_(k)≥2N, ∀k, it is possible to achieve the set of degrees offreedom tuples

(

):

$\begin{matrix}{{d_{j} = {M_{j}^{\prime}{\sum\limits_{m = 1}^{J}{( {T_{i_{1}} - {\sum\limits_{n = 1}^{m}M_{i_{n}}^{\prime}}} )( {\frac{1}{T_{i_{m}}} - \frac{1}{T_{i_{m + 1}}}} )}}}},} & (173)\end{matrix}$where M_(j)′=min{M_(j),[N−Σ_(m=1) ^(j−1), M_(k) _(m) ′]⁺}, and, fornotational convenience, introduce the trivial random variable T_(i)_(j+1) , i.e.,

$\frac{1}{T_{i_{J + 1}}} = 0.$Hence, the achievable degrees of freedom region is the convex hull ofthe degrees of freedom tuples,

(

), over all the

$\sum\limits_{i = 1}^{K}\frac{K!}{( {K - i} )!}$possible ordered set

⊆[1:K], i.e.,

={(d ₁ , . . . ,d _(K))∈Co(

(

),∀

⊆[1:K]}.  (174)

Proof:

By time-sharing between the transmission schemes that achieve thedegrees of freedom tuples

(

), we can construct the achievable degrees of freedom region which isthe convex hull of the achieved degrees of freedom tuples. The remainderof the proof is dedicated to show the achievability of the degrees offreedom tuple in Eq. (173) using the following transmission scheme.

-   -   Every T_(iJ)-length interval is divided into T_(i1)-length        subintervals.    -   During the first subinterval,        M_(j)′ pilots are sent to estimate M_(j)′ antennas of        transmitter j, and hence, during the following T_(i1)−        M_(j)′ period, the transmitters can communicate coherently        achieving M_(j)′(T_(i1)−        M_(j)′) degrees of freedom for transmitters j∈        .    -   During the remaining subintervals, when the index of the        subinterval is

${{\ell\;\frac{T_{i_{m}}}{T_{i_{1}}}} + 1},$where m=2, . . . , J−1, and

${\ell = 1},\ldots\mspace{14mu},{\frac{T_{i_{m} + 1}}{T_{i_{m}}} - 1},{\frac{T_{i_{m} + 1}}{T_{i_{m}}} + 1},\ldots\mspace{14mu},{{2\frac{T_{i_{m} + 1}}{T_{i_{m}}}} - 1},{\frac{T_{i_{m} + 1}}{T_{i_{m}}} + 1},\ldots\mspace{14mu},{\frac{T_{i_{j}}}{T_{i_{m}}} - 1},$the channel of transmitter j=i₁, . . . , i_(m) needs to be estimated,whereas the channel of transmitter j=i_(m)+1, . . . , J stays the same.Hence, Σ_(n=1) ^(m)M_(i) _(n) ′ pilots are sent to estimate M_(j)′antennas of transmitters j=i₁, . . . , i_(m). After that, during thefollowing T_(i1)−Σ_(n=1) ^(m)M_(i) _(n) ′ period, the transmitters cancommunicate coherently achieving M_(j)′(T_(i1)−Σ_(n=1) ^(m)M_(i) _(n) ′)degrees of freedom for transmitters j∈

. The number of subintervals of index

${k\;\frac{T_{m}}{T_{m - 1}}} + {1\mspace{14mu}{is}\mspace{14mu}{\sum\limits_{M = 2}^{J}{( {\frac{T_{i_{m + 1}}}{T_{i_{m}}} - 1} ){\frac{T_{i_{J}}}{T_{i_{m + 1}}}.}}}}$

-   -   For the T_(i1)-length subintervals whose index is not

${{\ell\;\frac{T_{i_{m}}}{T_{i_{1}}}} + 1},$the channels of all transmitters remain the same except the channel oftransmitter i₁. Hence, M_(i) ₁ ′ pilots are sent to estimate the channelof transmitter i₁, after that the transmitters can communicatecoherently during the following T_(i1)−M_(i) ₁ ′ period achievingM_(j)′(T_(i1)−M_(i) ₁ ′) degrees of freedom for transmitter j∈

. The number of the subintervals whose index is not equal to

${\ell\;\frac{T_{i_{m}}}{T_{i_{1}}}} + {1\mspace{14mu}{is}\mspace{14mu}( {\frac{T_{i_{2}}}{T_{i_{1}}} - 1} ){\frac{T_{i_{J}}}{T_{i_{2}}}.}}$For every T_(ij) interval transmitter j∈

achieves

$M_{j}^{\prime}{\sum\limits_{m = 1}^{J}{( {T_{i\; 1} - {\sum\limits_{n = 1}^{m}M_{i_{n}}^{\prime}}} )( {\frac{1}{T_{i_{m}}} - \frac{1}{T_{i_{m + 1}}}} )T_{i\; J}}}$degrees of freedom obtaining Eq. (173) which completes the proof ofTheorem 6.

B. Outer Bound

Theorem 7:

Consider a K-transmitter MIMO multiple access channel without CSIT orCSIR, meaning that the channel realization is not known, but the channeldistribution is globally known. Furthermore, the transmitters coherencetimes are assumed to be perfectly aligned and integer multiples of eachother. Define

={i₁, . . . , i_(J)}⊆[1:K] to be a set of J transmitters where

${\frac{T_{j}}{T_{j - 1}} \in},$∀j∈

, and T_(k)≥2 max_(k){M,N_(k)}, ∀k=1, . . . , K. For every

⊆[1:K], if a set of degrees of freedom tuples (d_(i1), . . . , d_(iJ))is achievable, then it must satisfy the inequalities

$\begin{matrix}{{\sum\limits_{j \in {\mathbb{J}}}d_{j}} \leq {\min\{ {N,{\sum\limits_{j \in {\mathbb{J}}}M_{j}}} \}{( {1 - \frac{\min\{ {N,{\sum\limits_{j \in {\mathbb{J}}}M_{j}}} \}}{T_{i_{J}}}} ).}}} & (175)\end{matrix}$

Proof:

The proof is divided into two parts. First, enhance the channel byincreasing the coherence times of the transmitters so that the enhancedchannel has identical coherence times.

Lemma 4:

For the considered K-transmitter MIMO multiple access channel, define

(

) to be the degrees of freedom region of a set of transmitters

={(i₁, . . . , i_(J)}⊆[1:K] with

${\frac{T_{j}}{T_{j - 1}} \in {\mathbb{Z}}},$∀j∈

. Define

(

) to be the degrees of freedom region of the same set of transmitters

={i₁, . . . , i_(J)}⊆[1:K] with T_(j)=T_(iJ), ∀j∈

, where the transmitters have identical coherence times, namely T_(iJ).Thus

(

)⊆

(

).  (176)The proof is provided below.

For the second part of the proof, the enhanced channel has identicalcoherence times, namely T_(iJ), hence, the cooperative outer bound is,

$\begin{matrix}{{\sum\limits_{j \in {\mathbb{J}}}R_{j}} \leq {I( {X(} }} & (177)\end{matrix}$According to the results of noncoherent communication, the bound in Eq.(175) can be obtained, and the proof of Theorem 7 is completed.

C. Numerical Examples

Consider a two-transmitter multiple access channel where thetransmitters are equipped with M₁=2, M₂=4 antennas and the receiver isequipped with N₂=4 antennas. The coherence time of the two transmittersis T₁=8 and T₂=32 slots. From Theorem 7, the outer degrees of freedomregion is given by

$\begin{matrix}\begin{matrix}{{d_{1} \leq \frac{3}{2}},} \\{{d_{1} + d_{2}} \leq {\frac{7}{2}.}}\end{matrix} & (178)\end{matrix}$The achievable degrees of freedom pairs in Theorem 6 can be obtained asfollows. For the case of two transmitters, there are 5 ordered sets oftransmitters

: { }, {1}, {2}, {1,2} and {2,1}. For { }, the trivial degrees offreedom pair (0,0) can be obtained. For the two sets {1}, {2}, thedegrees of freedom pairs

$( {\frac{3}{2},0} )$and

$( {0,\frac{7}{2}} ),$respectively, can be obtained. For the two sets {1,2} and {2,1}, thedegrees of freedom pairs

$( {\frac{11}{8},\frac{11}{8}} )$and

$( {0,\frac{7}{2}} ),$respectively, can be obtained. FIG. 24 shows an example of degrees offreedom region of a two-transmitter multiple access channel withheterogeneous coherence times where M₁=2, M₂=4, N=4, T₁=8, T₂=32. Theconvex hull of the achieved degrees of freedom pairs gives theachievable degrees of freedom region which is shown in FIG. 24.

Next, consider a two-transmitter multiple access channel where thetransmitters are equipped with M_(z)=3, M₂=2, N=3 antennas and coherencetimes T₁=8, T₂=24 slots. FIG. 25 shows an example of degrees of freedomregion of a two-transmitter multiple access channel with heterogeneouscoherence times where M₁=3, M₂=2, N=4, T=10, T₁=8, T₂=24. In this casethe achievable and the outer degrees of freedom regions are shown inFIG. 25.

VIII. Conclusion

In the second part of this disclosure, multi-user networks without CSITor CSIR were examined. For a broadcast channel where the receivers haveidentical coherence times, it was shown that the degrees of freedomregion is tight against the TDMA inner bound. However, when thereceivers have heterogeneous coherence times, TDMA is no longer optimalsince the difference of the coherence times can be a source of diversityin the wireless systems. For a broadcast channel where the receiver'scoherence times are integer multiples of each other, achievable degreesof freedom gains were obtained using the product superposition scheme.Furthermore, an outer degrees of freedom region was obtained usingchannel enhancement where the receiver's coherence times were increasedso that the receivers of the enhanced channel have identical coherencetimes. With the coherence time at least twice the number of transmit andreceive antennas, the optimality of the achievable scheme was shown infour cases: when the transmitter has fewer antennas than the receivers,when all the receivers have the same number of antennas, when thecoherence time of the receivers are very long compared to the coherencetime of one receiver, or the receivers have the same coherence time.

Additionally, a multiple access channel with identical coherence timeswas examined. A pilot-based achievable scheme was shown to be sumdegrees of freedom optimal. Furthermore, a multiple access channel withheterogeneous coherence times was considered. When the transmitterscoherence times are integer multiples of each other, an achievablepilot-based inner bound and an outer bound were obtained. The outerbound was obtained using channel enhancement where the transmitterscoherence times were increased so that the transmitters of the enhancedchannel have identical coherence times.

Coherent Broadcast Channel with Identical Coherent Times

The degrees of freedom optimality of TDMA inner bound (see, e.g.,paragraph 0079) can be shown when the receivers have identical coherencetimes and CSI is assumed to be available at the receiver. Enhance thechannel by providing global CSI at the receivers. Without loss ofgenerality, assume that N₁≤ . . . ≤N_(K). When M≤N₁, the cooperativeouter bound is tight against the TDMA inner bound. When M>N₁, thebroadcast channel is degraded, hence,

$\begin{matrix}\begin{matrix}{R_{i} \leq ( {U_{i}; Y_{i} |} } \\{{= {{I( {{X; Y_{i} \middle| {\mathbb{H}} },U^{i - 1}} )} - {I( {{X; Y_{i} \middle| {\mathbb{H}} },U^{i}} )}}},}\end{matrix} & (179)\end{matrix}$where U^(i)={U_(j)}_(j=1) ^(i) is a set of auxiliary random variablessuch that U₁→ . . . →U_(K−1)→X→(Y₁, . . . Y_(K)) forms a Markov chainand for notational convenience we introduced a trivial random variableU₀ and U_(K)=X.

is the set of all channels. Furthermore,R ₁≤(N ₁ −r ₁)log(ρ)+o(log(ρ)),R _(i) ≤I(X;Y _(i) |

,U ^(i−1))−r _(i) log(ρ)+o(log(ρ)),i≠1,K,R _(K) ≤I(X;Y _(K) |

,U ^(K−1)),  (180)since the degrees of freedom of I(X; Y₁|

) is bounded by the single-receiver bound, i.e. N₁, and r_(i) is definedto be the degrees of freedom of the term I(X; Y_(i)|

, U^(i)), where 0≤r_(i)≤N_(i)*. The extension of Lemma 1 of “On degreesof freedom region of MIMO networks without channel state information attransmitters” by C. Huang et al. (IEEE Trans. Inf. Theory, vol. 58, no.2, pp. 849-857, February 2012), which is hereby incorporated byreference in its entirety, to the K-receiver case is straight forward,and hence, can be written as

$\begin{matrix}{{{I( {{X; Y_{i,1} \middle| {\mathbb{H}} },U^{i},Y_{i,{2:N_{i}^{*}}}} )} \leq {{\frac{r_{i}}{N_{i}^{*}}{\log(\rho)}} + {o( {\log(\rho)} )}}},} & (181)\end{matrix}$where Y_(i,1)∈

^(1×T) is the received signal at antenna 1 of receiver i over the entireT-length coherence time whereas Y_(i,2:N) _(i) _(*) ∈

^((N) ^(i) ^(*) ^(−1)×T) is the matrix comprising the received signal atantennas 2,3, . . . , N_(i)* of receiver i over the entire T-lengthcoherence time. Furthermore,

$\begin{matrix}\begin{matrix}{{I( {{X; Y_{i} \middle| {\mathbb{H}} },U^{i - 1}} )}\overset{(a)}{=}{{I( {{X; Y_{i,{1:N_{i}^{*}}} \middle| {\mathbb{H}} },U^{i - 1}} )} +}} \\{I( {{X; Y_{i,{{N_{i}^{*} + 1}:N_{i}}} \middle| {\mathbb{H}} },U^{i - 1},Y_{i,{1:N_{i}^{*}}}} )} \\{\overset{(b)}{=}{{I( {{X; Y_{i,{1:N_{i\; - \; 1}^{*}}} \middle| {\mathbb{H}} },U^{i - 1}} )} +}} \\{{I( {{X; Y_{i,{{N_{i\; - \; 1}^{*} + 1}:N_{i}^{*}}} \middle| {\mathbb{H}} },U^{i - 1},Y_{i,{1:N_{i\; - \; 1}^{*}}}} )} +} \\{o( {\log(\rho)} )} \\{\overset{(c)}{=}{{I( {{X; Y_{{i - 1},{1:N_{i\; - 1}^{*}}} \middle| {\mathbb{H}} },U^{i - 1}} )} +}} \\{{I( {{X; Y_{i,{{N_{i\; - 1}^{*} + 1}:N_{i}^{*}}} \middle| {\mathbb{H}} },U^{i - 1},Y_{i,{1:N_{i\; - \; 1}^{*}}}} )} +} \\{o( {\log(\rho)} )} \\{= {{r_{i - 1}{\log(\rho)}} +}} \\{\sum\limits_{j = {N_{i - 1}^{*} + 1}}^{N_{i}^{*}}{I( {{X; Y_{i,j} \middle| {\mathbb{H}} },U^{i - 1},Y_{{i - 1},{1:N_{i - 1}^{*}}},} }} \\{ Y_{i,{{j + 1}:N_{i}^{*}}} ) + {o( {\log(\rho)} )}} \\{\overset{(d)}{\leq}{{r_{i - 1}{\log(\rho)}} +}} \\{{( {N_{i}^{*} - N_{i - 1}^{*}} ){I( {{X; Y_{{i - 1},1} \middle| {\mathbb{H}} },U^{i - 1},Y_{{i - 1},{2:N_{i - 1}^{*}}}} )}} +} \\{o( {\log(\rho)} )} \\{\overset{(e)}{\leq}{{r_{i - 1}{\log(\rho)}} + {( {N_{i}^{*} - N_{i - 1}^{*}} )\frac{r_{i - 1}}{N_{i - 1}^{*}}{\log(\rho)}} + {o( {\log(\rho)} )}}} \\{{\leq {{\frac{N_{i}^{*}}{N_{i - 1}^{*}}r_{i - 1}{\log(\rho)}} + {o( {\log(\rho)} )}}},}\end{matrix} & (182)\end{matrix}$where (a) and (b) follow from applying the chain rule, and h(Y_(i,N)_(i) _(*) _(+1:N) _(i) |

,U^(i−1),Y_(i,N) _(i) _(*) )=o(log(ρ)). Furthermore, (c) follows sinceY_(i,1:N) _(i−1) _(*) and Y_(i−1,1:N) _(i−1) _(*) , are statisticallythe same. (d) follows from applying the straight forward extension ofLemma 1 of “On degrees of freedom region of MIMO networks withoutchannel state information at transmitters” by C. Huang et al. (IEEETrans. Inf. Theory, vol. 58, no. 2, pp. 849-857, February 2012), whichis hereby incorporated by reference in its entirety, and (e) followsfrom Eq. (181). Therefore,

$\begin{matrix}{{d_{1} \leq {N_{1} - r_{1}}},} & (183) \\{{d_{i} \leq {{\frac{N_{i}^{*}}{N_{i - 1}^{*}}r_{i - 1}} - r_{i}}},{i \neq 1},K,} & \; \\{{d_{K} \leq {\frac{N_{K}^{*}}{N_{K - 1}^{*}}r_{K - 1}}},} & \;\end{matrix}$which gives the region defined in Eq. (120).

Proof of Lemma 1

First, we prove the equality of Eq. (122),

$\begin{matrix}\begin{matrix}{{I( {{X; {\overset{\_}{Y}}_{j} \middle| U },{\overset{\sim}{Y}}_{\{{j,\ell}\}}} )} = {{h( { {\overset{\_}{Y}}_{j} \middle| U ,{\overset{\sim}{Y}}_{\{{j,\ell}\}}} )} - {h( { {\overset{\_}{Y}}_{j} \middle| U ,{\overset{\sim}{Y}}_{\{{j,\ell}\}},X} )}}} \\{\overset{(a)}{=}{{h( { {\overset{\_}{Y}}_{\ell} \middle| U ,{\overset{\sim}{Y}}_{\{{j,\ell}\}}} )} - {h( { {\overset{\_}{Y}}_{\ell} \middle| U ,{\overset{\sim}{Y}}_{\{{j,\ell}\}},X} )}}} \\{= {{I( {{X; {\overset{\_}{Y}}_{\ell} \middle| U },{\overset{\sim}{Y}}_{\{{j,\ell}\}}} )}.}}\end{matrix} & (184)\end{matrix}$where (a) follows since the random variables are statisticallyequivalent, and entropies depend only on the statistics. Now prove theinequality of Eq. (123) as follows,

$\begin{matrix}\begin{matrix}{{I( {{X; {\overset{\_}{Y}}_{j} \middle| U },{\overset{\sim}{Y}}_{\{{j,\ell}\}}} )} = {{h( { {\overset{\_}{Y}}_{j} \middle| U ,{\overset{\sim}{Y}}_{\{{j,\ell}\}}} )} - {h( { {\overset{\_}{Y}}_{j} \middle| U ,{\overset{\sim}{Y}}_{\{{j,\ell}\}},X} )}}} \\{\overset{(a)}{\geq}{{h( { {\overset{\_}{Y}}_{j} \middle| U ,{\overset{\sim}{Y}}_{\{{j,\ell}\}},{\overset{\_}{Y}}_{\ell}} )} - {h( { {\overset{\_}{Y}}_{j} \middle| U ,{\overset{\sim}{Y}}_{\{{j,\ell}\}},X} )}}} \\{\overset{(b)}{=}{{h( { {\overset{\_}{Y}}_{j} \middle| U ,{\overset{\sim}{Y}}_{\{{j,\ell}\}},{\overset{\_}{Y}}_{\ell}} )} - {h( { {\overset{\_}{Y}}_{j} \middle| U ,{\overset{\sim}{Y}}_{\{{j,\ell}\}},X,{\overset{\_}{Y}}_{\ell}} )}}} \\{{= {I( {{X; {\overset{\_}{Y}}_{j} \middle| U },{\overset{\sim}{Y}}_{\{{j,\ell}\}},{\overset{\_}{Y}}_{\ell}} )}},}\end{matrix} & (185)\end{matrix}$where (a) follows since conditioning does not increase the entropy.Furthermore (b) follows by the fact that Y _(j)→X→Y _(l) forms a Markovchain. The received signal Y _(j), Y _(l) are given by, respectively,Y _(T) =h _(j) ^(H) X+z _(j) ^(H),Y _(l) =h _(l) ^(H) X+z _(l) ^(H),  (186)where h_(j) ^(H), h_(l) ^(H) are the channel vectors which areindependent from each other whereas z_(j) ^(H), z_(l) ^(H) are theindependent corresponding noise vectors. Therefore, conditioning on X, Y_(j) and Y _(l) are independent.

Proof of Lemma 2

Consider the set of receivers

⊆[1:K] where the receivers are ordered ascendingly according to thecoherence time length, i.e., T_(j)≥T_(j), ∀j∈

. The proof consists of two steps. First, show that the individualdegrees of freedom of each receiver is nondecreasing with the increaseof the coherence time of this receiver. Second, show that the degrees offreedom region of the channel is nondecreasing with the increase of thecoherence time of the receivers. For the first step of the proof,introduce the following Lemma.

Lemma 5:

For the broadcast channel considered in Section III, define

={i₁, . . . , i_(J)}⊆[1:K] with T_(j)≥T_(j−1), ∀j∈

and Ψ as the message of receive j∈

r. Thus,

$\begin{matrix}{{{N_{j}^{*}( {\frac{1}{J} - \frac{N_{j}^{*}}{T_{j}}} )} \leq {{MG}\{ {\frac{1}{\overset{\_}{n}}{I( {\Psi_{j};Y_{j}^{n}} )}} \}} \leq {N_{j}^{*}( {1 - \frac{N_{j}^{*}}{T_{j}}} )}},} & (187)\end{matrix}$where MG(x) is the multiplexing gain of a function x(ρ) of ρ and definedas

$\begin{matrix}{{{MG}(x)} = {\lim\limits_{\rhoarrow\infty}{\sup{\frac{x(\rho)}{\log(\rho)}.}}}} & (188)\end{matrix}$

Proof: To first prove the right inequality of Eq. (187)

$\begin{matrix}\begin{matrix}{{{MG}\{ {\frac{1}{\overset{\_}{n}}{I( {\Psi_{j};Y_{j}^{n}} )}} \}}\overset{(a)}{\leq}{{MG}\{ {\frac{1}{n}{I( {X^{n};Y_{j}^{n}} )}} \}}} \\{{\overset{(b)}{\leq}{N_{j}^{*}( {1 - \frac{N_{j}^{*}}{T_{j}}} )}},}\end{matrix} & (189)\end{matrix}$where (a) follows from the data processing inequality and (b) followsfrom the single-receiver results. Next, we show the left inequality ofEq. (187). Assume the following transmitted sequenceX ^(n) U _(i) ₁ , . . . ,U _(i) _(J) ],  (190)where

$U_{j} \in {\mathbb{C}}^{N_{j} \times \frac{\overset{\_}{n}}{j}}$is the matrix containing the signal of receiver j∈

and the matrix is constructed to be on the form of the optimal input ofa non-coherent single receiver. Hence,

$\begin{matrix}{{{MG}\{ {\frac{1}{n}{I( {\Psi_{j};Y_{j}^{n}} )}} \}} \geq {{MG}\{ {\frac{1}{n}{I( {U_{j};Y_{j}^{n}} )}} \}} \geq {\frac{N_{j}^{*}}{J}( {1 - \frac{N_{j}^{*}}{T_{j}}} )} \geq {{N_{j}^{*}( {\frac{1}{J} - \frac{N_{j}^{*}}{T_{j}}} )}.}} & (191)\end{matrix}$Thus, the proof of Lemma 5 is completed.

By Lemma 5, there are lower and outer bounds which are increasing withT_(j). Furthermore, the difference between the two bounds is

$\begin{matrix}{\Delta = {{{N_{j}^{*}( {1 - \frac{N_{j}^{*}}{T_{j}}} )} - {N_{j}^{*}( {\frac{1}{J} - \frac{N_{j}^{*}}{T_{j}}} )}} = {N_{j}^{*}{\frac{J - 1}{J}.}}}} & (192)\end{matrix}$Therefore,

${MG}\{ {\frac{1}{n}{I( {\Psi_{j};Y_{j}^{n}} )}} \}$is nondecreasing with the increase of T_(j), which completes the firststep of the proof of Lemma 2.

Now, consider the second part of the proof via a contradiction argument.Define

to be the degrees of freedom region of a set of receivers with unequalcoherence times where max_(j)T_(j)=T_(max) and Y_(j) ^(n) denotes thereceived signal at receiver j. Define

to be the degrees of freedom region of the receivers where the coherencetime of all receivers is T_(max), where Y _(j) ^(n) denotes the receivedsignal at receiver j of this enhanced channel. Define {tilde over (D)}∈

to be a degrees of freedom tuple, and d_(j)∈{tilde over (D)} is thedegrees of freedom of receiver j. Assume that {tilde over (D)}∉

. By Fano's inequality,

$\begin{matrix}\begin{matrix}{{d_{j} \leq {{MG}\{ {\frac{1}{\overset{\_}{n}}{I( {\Psi_{j};Y_{j}^{n}} )}} \}}},} \\{{\leq {{MG}\{ {\frac{1}{\overset{\_}{n}}{I( {\Psi_{j};{\overset{\_}{Y}}_{j}^{n}} )}} \}}},}\end{matrix} & (193)\end{matrix}$where Ψ_(j) is the message of receiver j∈

, and the last inequality follows from Lemma 5. Therefore, d_(j)∈

, ∀j, which contradicts the initial assumption completing the secondpart of the proof.

Proof of Lemma 4

Consider the set of transmitter

={i₁, . . . , i_(J)}⊆[1:K] where ∀j∈

,

$\frac{T_{j}}{T_{j - 1}} \in .$By Fano's inequality, as n→∞,

$\begin{matrix}{{{R_{j}} \leq {\frac{1}{\overset{\_}{n}}{I( {X_{i_{1}}^{n},\ldots\mspace{14mu},{X_{i_{J}}^{n};Y^{n}}} )}}},} & (194)\end{matrix}$where X_(j) ^(n) is transmitter j∈

signal and Y^(n) is the received signal over the entire transmissiontime 1:n. In the sequel, it can be shown that the degrees of freedom of

${\frac{1}{n}{I( {X_{j};Y^{n}} )}},$j∈

is nondecreasing in T_(j). Given lower and upper bounds on this term,and furthermore, both bounds are nondecreasing in T_(j). Introduce thefollowing Lemma.

Lemma 6:

For the multiple access channel considered in Section VII, define

={i₁, . . . , i_(J)}⊆[1:K] and Ψ_(j) as the message of transmitter j∈

. Thus,

$\begin{matrix}{{{M_{j}^{*}( {\frac{1}{J} - \frac{M_{j}^{*}}{T_{j}}} )} \leq {{MG}\{ {\frac{1}{\overset{\_}{n}}{I( {X_{i_{1}}^{n},\ldots\mspace{14mu},{X_{i_{J}}^{n};Y^{n}}} )}} \}} \leq {M_{j}^{*}( {1 - \frac{M_{j}^{*}}{T_{j}}} )}},} & (195)\end{matrix}$Proof:

First prove the right inequality of Eq. (195). Begin with

$\begin{matrix}{{{{MG}\{ {\frac{1}{\overset{\_}{n}}{I( {X_{i_{1}}^{n},\ldots\mspace{14mu},{X_{i_{J}}^{n};Y^{n}}} )}} \}} \leq {M_{j}^{*}( {1 - \frac{M_{j}^{*}}{T_{j}}} )}},} & (196)\end{matrix}$where the above inequality follows from the single-transmitter results.Next, show the left inequality of Eq. (195). Assume the followingtransmitted sequenceX _(j) ^(n)=[0, . . . 0,U _(j),0, . . . ,0],  (197)where U_(j)∈

M j × n _ Jis the matrix containing the signal of transmitter j∈

and the matrix is constructed to be on the form of the optimal input ofa non-coherent single transmitter. Hence,

$\begin{matrix}{{{MG}\{ {\frac{1}{\overset{\_}{n}}{I( {X_{i_{1}}^{n},\ldots\mspace{14mu},{X_{i_{J}}^{n};Y^{n}}} )}} \}} \geq {{MG}\{ {\frac{1}{\overset{\_}{n}}{I( {U_{j};Y_{j}^{n}} )}} \}} \geq {\frac{M_{j}^{*}}{J}( {1 - \frac{M_{j}^{*}}{T_{j}}} )} \geq {{M_{j}^{*}( {\frac{1}{J} - \frac{M_{j}^{*}}{T_{j}}} )}.}} & (198)\end{matrix}$Thus, the proof of Lemma 6 is completed.

By Lemma 6, there exists lower and outer bounds which are increasingwith T_(j). Furthermore, the difference between the two bounds is

$\begin{matrix}{\Delta = {{{M_{j}^{*}( {1 - \frac{M_{j}^{*}}{T_{j}}} )} - {M_{j}^{*}( {\frac{1}{J} - \frac{M_{j}^{*}}{T_{j}}} )}} = {M_{j}^{*}{\frac{J - 1}{J}.}}}} & (199)\end{matrix}$Therefore

${MG}\{ {\frac{1}{n}{I( {X_{i_{1}}^{n},\ldots\mspace{14mu},{X_{i_{J}}^{n};Y^{n}}} )}} \}$is nondecreasing with the increase of T_(j), and hence, the proof ofLemma 4 is completed.

Referring next to FIG. 26, shown is a flow chart illustrating an exampleof the coherence diversity which can be implemented by processingcircuitry in transceivers, transmitters and/or receivers, in accordancewith various embodiments of the present disclosure. Consider the caseof, e.g., a transceiver. Beginning at 2603, a product signal transmittedover a plurality of subcarriers is received. The product signal cancomprise a product superposition of a first baseband signal and a secondbaseband signal. The first baseband signal can include a pilot symbol ina number of time slots of at least a portion of the plurality ofsubcarriers and a first encoded message in a remaining number of timeslots of the plurality of subcarriers. The second baseband signal caninclude a second encoded message. In other embodiments, the productsignal can comprise a product superposition of three or more basebandsignals. For example, the product signal can comprise a productsuperposition of the first baseband signal, the second baseband signaland a third baseband signal.

The number of time slots can correspond to a number of antennastransmitting the product signal. The number of time slots can be abeginning number of time slots of the portion of the plurality ofsubcarriers. In other embodiments, the number of time slots may notinclude a beginning number of time slots of the portion of the pluralityof subcarriers. In some implementations, each of the plurality ofsubcarriers can include the pilot symbol in the number of time slots. Inother implementations, a first number of the plurality of subcarrierscan include the pilot symbol in the number of time slots, and aremaining number of the plurality of subcarriers will not include thepilot symbol. For example, one subcarrier can include the pilot symboland the remaining subcarriers do not include the pilot symbol. The pilotsymbol can be included in a subgroup of the number of time slots.Individual time slots of the subgroup can be separated based upon thefirst coherence time. Time slots between the individual time slots ofthe subgroup can comprise a portion of the first encoded message. Thesubgroup of the number of time slots can be based upon a number ofantennas transmitting the product signal, a number of antennas receivingthe product signal, and the first coherence time.

The first baseband signal can be associated with a first channel thatvaries faster in time than a second channel associated with the secondbaseband signal. The first baseband signal can be associated with afirst channel that varies faster in both time and frequency than asecond channel associated with the second baseband signal. In someembodiments, the second channel can vary faster in frequency than thefirst channel. In other implementations, the first baseband signal canbe associated with a first channel that varies faster in frequency thana second channel associated with the second baseband signal. The secondchannel can vary faster in time than the first channel.

The equivalent channel responses for the subcarriers can be estimatedbased upon the pilot symbol in the number of time slots of thesubcarriers at 2606. The equivalent channel responses for the firstnumber of the plurality of subcarriers can be estimated based upon thepilot symbol in the number of time slots of the first number of theplurality of subcarriers, and the equivalent channel responses for theremaining number of the plurality of subcarriers can be interpolatedbased upon the pilot symbol in the number of time slots of the firstnumber of the plurality of subcarriers. The number of time slots and theremaining number of time slots can correspond to subintervals of a firstcoherence time interval of a first receiver of the first baseband signaland a second coherence time interval of a second receiver of the secondbaseband signal. In some embodiments, the number of time slots and theremaining number of time slots can correspond to subintervals of a firstcoherence time interval of a first receiver of the first basebandsignal, a second coherence time interval of a second receiver of thesecond baseband signal, and a third coherence time interval of a thirdreceiver of the second baseband signal. The second and third coherencetimes can be integer multiples or non-integer multiples of the firstcoherence time. Interference alignment of the first and second basebandsignals can be included after estimating the equivalent channelresponses.

Encoded messages can then be decoded at 2609. The second encoded messagecan be decoded based at least in part upon the first baseband signal andthe equivalent channel responses. Decoding the second encoded messagecan include removing the first baseband signal from the product signalbased upon the equivalent channel responses. In other implementations,decoding the second encoded message can comprise estimating the secondbaseband signal from the product of the pilot symbol and the secondencoded message.

With reference now to FIG. 27, shown is a schematic block diagram of anexample of processing circuitry 2700 in transceivers, transmittersand/or receivers that may be used to implement various portions of thecoherency diversity in accordance with various embodiments of thepresent disclosure. The processing circuitry 2700 includes at least oneprocessor circuit, for example, having a processor 2703 and a memory2706, both of which are coupled to a local interface 2709. To this end,the processing circuitry 2700 may be implemented using one or morecircuits, one or more microprocessors, microcontrollers, applicationspecific integrated circuits, dedicated hardware, digital signalprocessors, microcomputers, central processing units, field programmablegate arrays, programmable logic devices, state machines, or anycombination thereof. The local interface 2709 may comprise, for example,a data bus with an accompanying address/control bus or other busstructure as can be appreciated. The processing circuitry 2700 caninclude a display for rendering of generated graphics such as, e.g., auser interface and an input interface such, e.g., a keypad or touchscreen to allow for user input. In addition, the processing circuitry2700 can include communication interfaces (not shown) that allow theprocessing circuitry 2700 to communicatively couple with othercommunication devices. The communication interfaces may include one ormore wireless connection(s) such as, e.g., Bluetooth or other radiofrequency (RF) connection and/or one or more wired connection(s).

Stored in the memory 2706 are both data and several components that areexecutable by the processor 2703. In particular, stored in the memory2706 and executable by the processor 2703 are coherence diversityapplication(s) 2715, an operating system 2718, and/or other applications2721. Coherence diversity applications 2715 can include applicationsthat support control and/or operation of the transceiver, transmitterand/or receiver for communications. It is understood that there may beother applications that are stored in the memory 2706 and are executableby the processor 2703 as can be appreciated. Where any componentdiscussed herein is implemented in the form of software, any one of anumber of programming languages may be employed such as, for example, C,C++, C #, Objective C, Java®, JavaScript®, Perl, PHP, Visual Basic®,Python, Ruby, Delphi®, Flash®, LabVIEV® or other programming languages.

A number of software components are stored in the memory 2706 and areexecutable by the processor 2703. In this respect, the term “executable”means a program file that is in a form that can ultimately be run by theprocessor 2703. Examples of executable programs may be, for example, acompiled program that can be translated into machine code in a formatthat can be loaded into a random access portion of the memory 2706 andrun by the processor 2703, source code that may be expressed in properformat such as object code that is capable of being loaded into a randomaccess portion of the memory 2706 and executed by the processor 2703, orsource code that may be interpreted by another executable program togenerate instructions in a random access portion of the memory 2706 tobe executed by the processor 2703, etc. An executable program may bestored in any portion or component of the memory 2706 including, forexample, random access memory (RAM), read-only memory (ROM), hard drive,solid-state drive, USB flash drive, memory card, optical disc such ascompact disc (CD) or digital versatile disc (DVD), floppy disk, magnetictape, or other memory components.

The memory 2706 is defined herein as including both volatile andnonvolatile memory and data storage components. Volatile components arethose that do not retain data values upon loss of power. Nonvolatilecomponents are those that retain data upon a loss of power. Thus, thememory 2706 may comprise, for example, random access memory (RAM),read-only memory (ROM), hard disk drives, solid-state drives, USB flashdrives, memory cards accessed via a memory card reader, floppy disksaccessed via an associated floppy disk drive, optical discs accessed viaan optical disc drive, magnetic tapes accessed via an appropriate tapedrive, and/or other memory components, or a combination of any two ormore of these memory components. In addition, the RAM may comprise, forexample, static random access memory (SRAM), dynamic random accessmemory (DRAM), or magnetic random access memory (MRAM) and other suchdevices. The ROM may comprise, for example, a programmable read-onlymemory (PROM), an erasable programmable read-only memory (EPROM), anelectrically erasable programmable read-only memory (EEPROM), or otherlike memory device.

Also, the processor 2703 may represent multiple processors 2703 and thememory 2706 may represent multiple memories 2706 that operate inparallel processing circuits, respectively. In such a case, the localinterface 2709 may be an appropriate network that facilitatescommunication between any two of the multiple processors 2703, betweenany processor 2703 and any of the memories 2706, or between any two ofthe memories 2706, etc. The local interface 2709 may comprise additionalsystems designed to coordinate this communication, including, forexample, performing load balancing. The processor 2703 may be ofelectrical or of some other available construction.

Although the coherence diversity application(s) 2715, the operatingsystem 2718, application(s) 2721, and other various systems describedherein may be embodied in software or code executed by general purposehardware as discussed above, as an alternative the same may also beembodied in dedicated hardware or a combination of software/generalpurpose hardware and dedicated hardware. If embodied in dedicatedhardware, each can be implemented as a circuit or state machine thatemploys any one of or a combination of a number of technologies. Thesetechnologies may include, but are not limited to, discrete logiccircuits having logic gates for implementing various logic functionsupon an application of one or more data signals, application specificintegrated circuits having appropriate logic gates, or other components,etc. Such technologies are generally well known by those skilled in theart and, consequently, are not described in detail herein.

Also, any logic or application described herein, including the coherencediversity application(s) 2715 and/or application(s) 2721, that comprisessoftware or code can be embodied in any non-transitory computer-readablemedium for use by or in connection with an instruction execution systemsuch as, for example, a processor 2703 in a computer system or othersystem. In this sense, the logic may comprise, for example, statementsincluding instructions and declarations that can be fetched from thecomputer-readable medium and executed by the instruction executionsystem. In the context of the present disclosure, a “computer-readablemedium” can be any medium that can contain, store, or maintain the logicor application described herein for use by or in connection with theinstruction execution system. The computer-readable medium can compriseany one of many physical media such as, for example, magnetic, optical,or semiconductor media. More specific examples of a suitablecomputer-readable medium would include, but are not limited to, magnetictapes, magnetic floppy diskettes, magnetic hard drives, memory cards,solid-state drives, USB flash drives, or optical discs. Also, thecomputer-readable medium may be a random access memory (RAM) including,for example, static random access memory (SRAM) and dynamic randomaccess memory (DRAM), or magnetic random access memory (MRAM). Inaddition, the computer-readable medium may be a read-only memory (ROM),a programmable read-only memory (PROM), an erasable programmableread-only memory (EPROM), an electrically erasable programmableread-only memory (EEPROM), or other type of memory device.

It should be emphasized that the above-described embodiments of thepresent disclosure are merely possible examples of implementations setforth for a clear understanding of the principles of the disclosure.Many variations and modifications may be made to the above-describedembodiment(s) without departing substantially from the spirit andprinciples of the disclosure. All such modifications and variations areintended to be included herein within the scope of this disclosure andprotected by the following claims.

It should be noted that ratios, concentrations, amounts, and othernumerical data may be expressed herein in a range format. It is to beunderstood that such a range format is used for convenience and brevity,and thus, should be interpreted in a flexible manner to include not onlythe numerical values explicitly recited as the limits of the range, butalso to include all the individual numerical values or sub-rangesencompassed within that range as if each numerical value and sub-rangeis explicitly recited. To illustrate, a concentration range of “about0.1% to about 5%” should be interpreted to include not only theexplicitly recited concentration of about 0.1% to about 5%, but alsoinclude individual concentrations (e.g., 1%, 2%, 3%, and 4%) and thesub-ranges (e.g., 0.5%, 1.1%, 2.2%, 3.3%, and 4.4%) within the indicatedrange. The term “about” can include traditional rounding according tosignificant figures of numerical values. In addition, the phrase “about‘x’ to ‘y’” includes “about ‘x’ to about ‘y’”.

Therefore, at least the following is claimed:
 1. A method, comprising:receiving a product signal transmitted over a plurality of subcarriers,the product signal comprising a product superposition of a firstbaseband signal and a second baseband signal, where the first basebandsignal comprises a pilot symbol in a number of time slots of at least aportion of the plurality of subcarriers and a first encoded message in aremaining number of time slots of the plurality of subcarriers, where afirst number of the plurality of subcarriers includes the pilot symbolin the number of time slots, and a remaining number of the plurality ofsubcarriers do not include the pilot symbol, and where the secondbaseband signal comprises a second encoded message; estimatingequivalent channel responses for the plurality of subcarriers based uponthe pilot symbol in the number of time slots of the plurality ofsubcarriers, wherein the equivalent channel responses for the firstnumber of the plurality of subcarriers are estimated based upon thepilot symbol in the number of time slots of the first number of theplurality of subcarriers, and the equivalent channel responses for theremaining number of the plurality of subcarriers are interpolated basedupon the pilot symbol in the number of time slots of the first number ofthe plurality of subcarriers; and decoding the second encoded messagebased at least in part upon the first baseband signal and the equivalentchannel responses.
 2. The method of claim 1, wherein the number of timeslots are a beginning number of time slots of the at least a portion ofthe plurality of subcarriers.
 3. The method of claim 1, wherein decodingthe second encoded message comprises removing the first baseband signalfrom the product signal based upon the equivalent channel responses. 4.The method of claim 1, wherein decoding the second encoded messagecomprises estimating the second baseband signal from the product of thepilot symbol and the second encoded message.
 5. The method of claim 1,wherein the number of time slots corresponds to a number of antennastransmitting the product signal.
 6. The method of claim 1, wherein thefirst number of the plurality of subcarriers is one subcarrier of theplurality of subcarriers.
 7. The method of claim 1, wherein the numberof time slots and the remaining number of time slots correspond tosubintervals of a first coherence time interval of a first receiver ofthe first baseband signal and a second coherence time interval of asecond receiver of the second baseband signal.
 8. The method of claim 1,wherein the product signal comprises a product superposition of three ormore baseband signals.
 9. The method of claim 1, wherein the productsignal comprises a product superposition of the first baseband signal,the second baseband signal and a third baseband signal.
 10. The methodof claim 1, wherein the first baseband signal is associated with a firstchannel that varies faster in time than a second channel associated withthe second baseband signal.
 11. The method of claim 10, wherein thefirst baseband signal is associated with a first channel that variesfaster in both time and frequency than a second channel associated withthe second baseband signal.
 12. The method of claim 10, wherein thesecond channel varies faster in frequency than the first channel. 13.The method of claim 1, wherein the first baseband signal is associatedwith a first channel that varies faster in frequency than a secondchannel associated with the second baseband signal.
 14. The method ofclaim 13, wherein the second channel varies faster in time than thefirst channel.
 15. The method of claim 1, further comprisinginterference alignment of the first and second baseband signals afterestimating the equivalent channel responses.
 16. A method, comprising:receiving a product signal transmitted over a plurality of subcarriers,the product signal comprising a product superposition of a firstbaseband signal, a second base band signal and a third baseband signal,where the first baseband signal comprises a pilot symbol in a number oftime slots of at least a portion of the plurality of subcarriers and afirst encoded message in a remaining number of time slots of theplurality of subcarriers, and where the second baseband signal comprisesa second encoded message, wherein the number of time slots and theremaining number of time slots correspond to subintervals of a firstcoherence time interval of a first receiver of the first basebandsignal, a second coherence time interval of a second receiver of thesecond baseband signal, and a third coherence time interval of a thirdreceiver of the second baseband signal; estimating equivalent channelresponses for the plurality of subcarriers based upon the pilot symbolin the number of time slots of the plurality of subcarriers; anddecoding the second encoded message based at least in part upon thefirst baseband signal and the equivalent channel responses.
 17. Themethod of claim 16, wherein the number of time slots do not include abeginning number of time slots of the at least a portion of theplurality of subcarriers.
 18. The method of claim 16, wherein each ofthe plurality of subcarriers includes the pilot symbol in the number oftime slots.
 19. The method of claim 16, wherein a first number of theplurality of subcarriers includes the pilot symbol in the number of timeslots, and a remaining number of the plurality of subcarriers do notinclude the pilot symbol.
 20. The method of claim 16, wherein second andthird coherence times are integer multiples of a first coherence time.21. The method of claim 16, wherein the second and third coherence timesare non-integer multiples of the first coherence time.
 22. The method ofclaim 16, wherein the pilot symbol is included in a subgroup of thenumber of time slots.
 23. The method of claim 22, wherein individualtime slots of the subgroup are separated based upon a first coherencetime.
 24. The method of claim 23, wherein time slots between theindividual time slots of the subgroup comprise a portion of the firstencoded message.
 25. The method of claim 22, wherein the subgroup of thenumber of time slots is based upon a number of antennas transmitting theproduct signal, a number of antennas receiving the product signal, andthe first coherence time.